Complex Symmetries 3030880583, 9783030880583

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György Darvas Editor

Complex Symmetries

György Darvas Editor

Complex Symmetries

Editor György Darvas Symmetrion Budapest, Hungary

ISBN 978-3-030-88058-3 ISBN 978-3-030-88059-0 (eBook) https://doi.org/10.1007/978-3-030-88059-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

COMPLEX SYMMETRIES CONTENTS 

Complex Symmetries, Introduction, György Darvas

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Complex Symmetries in Repeating Hyperbolic Patterns, Douglas Dunham

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IgnoTheory: A Compositional System for Intermedia Art Based on Tiling Patterns and Labelled Graphs, Paul Hertz

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Symmetries of Maps on Surfaces, Ashish Kumar Upadhyay

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Symmetry in Projection of 4-Dimensional Regular Polychora, Koji Miyazaki and Motonaga Ishii 43

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Morphic Polytopes and Symmetries, Guy Inchbald

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Inducing the Symmetries Out of the Complexity: The Kepler Triangle and Its Kin as a Model Problem, Takeshi Sugimoto

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Synchronizing the Isotropic Vector Matrix with the Stellated Vector Matrix, Jim Lehman 79

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An Analogy and Several Symmetries, András Recski

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Discrete Lattices on the Single Bearing Spiral: From Geometry to Botany, Dmitriy Gurevich 105

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Symmetry and Invariance: Interdisciplinary Teaching, Simone Brasili and Riccardo Piergallini 123

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Dilative Rotation, Dilative Reflection in Mathematics, Nature, Art, and Education, Eleonóra Stettner, György Emese

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Relationship of Symmetry and Combinatorics in the Poly-Universe Game, János Szász SAXON, Gábor Kis 163

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Symmetry in Arrow-Like Salt Crystals, Marina Voinova

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Symmetries in Stellar, Galactic, and Extragalactic Astronomy, László Szabados

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Symmetries and the Genetic Code, Sergey V. Petoukhov, Elena S. Petukhova, Vladimir V. Verevkin 207 v

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Global Symmetry Local Asymmetry: In the Realized Buildings by the Innovation Studio ONL BV, Kas Oosterhuis

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Theme, Motive, Structure and Symmetry – Pentasonata by Andrzej Panufnik, Katarzyna Szymańska-Stułka

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Fluid Symmetry: Logical to Artists, Mesmerizing to Viewers, Fré Ilgen

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Complex Symmetries in Réograms, Paul B. Ré

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______________ List of graphic illustrations of complex symmetries Tamás F. Farkas Patrice Jeener

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COMPLEX SYMMETRIES György Darvas

Physicist, philosopher (b. Budapest, Hungary, 1948). Symmetrion, Budapest, https://symmetry.hu/symmetrion/. E-mail: [email protected]; web: http://isa.symmetry.hu/members/darvasg/

Many books were published on symmetries in the recent few decades. The majority dealt with phenomena, things that display some appearance of symmetry in the sciences and arts. There are simple and complex phenomena among them. Only a few discussed the nature of symmetries as such. Symmetries appear in simple and complex forms. Both simple and complex phenomena can show simple and complex symmetries either. Why has been the nature of symmetries less discussed? According to my experience, in most cases, authors do not consider important to explain why this or that subject of their studies is symmetric. They have worked with those phenomena and concepts for long and the symmetry properties of those seem obvious, at least for them. Therefore, most authors ignore explaining why is their subject symmetric. This is acceptable for colleagues active in the same discipline. The same does not hold for the general reader. Everybody has a certain idea of what is symmetry. However, the associations that the word awakens in a common person depend on the experiences that have established its meaning for her/him. The term ‘symmetry’ can have three separate types of meaning, as a phenomenon, a concept or an operation. The phenomenon is what we consider to be symmetrical based on our experience or of knowledge we have obtained. The concept is what circumscribes all such phenomena. The operation is what gives rise to the phenomenon or makes it possible. The meaning, and even the mutual relation of these words, seem too much scientific for the average person. Everyone will more likely enjoy or appreciate when meets something symmetric if s/he is aware what makes that thing symmetric and what is the nature of that symmetry. This is the goal of the investigations into symmetries what is called symmetrology.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Darvas (ed.), Complex Symmetries, https://doi.org/10.1007/978-3-030-88059-0_1

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Symmetrology has not been built in traditional curricula at university education. Only crystallography included analysing the nature of symmetries in its teaching program. Other disciplines discuss either partially (e.g., physics) or not a bit the nature of the symmetries of their subjects. Symmetrology became part of the curricula at a few but an increasing number of universities only in the recent two decades. Even in these cases, this occurs in general terms, that means, phenomenological aspects are studied together with the analysis of the nature of the respective symmetries. Let us begin acquainting the common concept of symmetry with the ordinary meaning of the word and built up its academically-founded concept. On hearing the word symmetry, the majority of people think of the simplest examples of the concept. The best-known example of geometric symmetry is reflection. Do you all know exactly what is this? In precise terms, if we reflect a (planar) shape in a linear one (the axis of symmetry), then it appears on the far side such that the respective points of the shape and its reflection are at the same distance from the axis, albeit in opposite directions. The figure retains its shape – in mirrored form – and its size and the angles between the lines connecting its various points are also unchanged, as is its colour. Reflection changes the direction of orientation, however: left and right are swapped. We have completed an operation, that of reflection, in the course of which certain characteristics of the reflected object have changed, but some have remained the same. It is these unchanged characteristics that represent the symmetry of the original figure in relation to its reflection. This is a discrete symmetry because if we reflect the reflected image in the same axis again, we are returned to the original shape. In two steps we can return to the original arrangement. Taking the same reflection operation one dimension further, we can produce the same results with spatial figures by using a mirror plane instead of an axis. There are three-dimensional mirror images which cannot be made to overlap each other in space by any kind of geometric operation: our left and right hands are examples. Based on this (left- or right-) handedness, such figures are referred to as chiral (or chiral symmetric) from the Greek. The second-most mentioned symmetry operation is rotation. If we rotate a (planar) figure around an axis perpendicular to the plane, the figure preserves its internal characteristics and the distance of its points from the axis. Its symmetry lies in keeping these properties intact. If we complete a rotation of such an angle that after a finite number of rotations – 2, 3, 4, 5, 6, .... – the figure exactly overlaps the original, then, depending on the angle of the rotation, we can talk of 2-, 3-, 4-, 5- or 6-fold symmetry. In general, if we rotate the

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object by an angle of 360º/n, we term this n-fold symmetry, where n is a natural number. n-fold symmetry is also a discrete symmetry because after n identical steps we get back to the original. We can rotate spatial figures in the same way around an axis. The operation we have completed is a rotation around an axis. The object on which we completed it is a geometric figure. Except for its spatial position, almost all of its properties have remained intact, whatever we take our frame of reference to be. Less commonly known than the above two in everyday thinking, but still regularly mentioned, is translational symmetry. If we translate a (planar) figure along a straight line with a uniform period, always in the same direction, we are given a repeated series of the given figure, in which the repeated elements are alike in every respect, and the distance between which is fixed on a periodic basis. Their symmetry lies in this uniformity. Translational symmetry is characterized by the direction of the translation and the length of the period. The symmetry operation in this instance is the straight linear shift (translation) we perform on a geometric object, as in the previous instances; in the process of this operation, its all characteristics are preserved, save its spatial location. It is in this form that symmetry most often occurs in our surroundings. This is true of the windows on most buildings, the successive carriages of trains, series of telegraph poles and electricity pylons, bars of railings, paving stones on the street, the web of squares in a school notebook, the rows of chairs in a schoolroom or theatre, the headstones in a cemetery, the frieze designs on Greek vases, the atoms in crystals, the waves of a lake, the annual recurrence of the changing seasons, the phases of the Moon, the rhythm of songs, dance movements, and the rhyme of poems. Reflection, rotation and translation are all simple symmetries. They involve a single symmetry operation each. One could list a few further simple symmetries (like similitude, affine projection, topological symmetry, conjugation, …). Nevertheless, these three simple geometric operations exemplify simple geometric symmetries. Complex symmetries are combinations of simple symmetries. Let us present an easy example of a complex symmetry. In decorative designs (freeze designs) we often encounter glide reflection. In essence, this is a combined (complex) symmetry operation. The object to be reflected is first glided by one-period length, then reflected in an axis parallel with the direction of gliding, and so on. That means, glide reflection is composed of a translation and a reflection operation. The preserved characteristics are made up of the characteristics of the reflection and the glide symmetry, but each pair of elements can also be interpreted as a translational symmetry. Given that,

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in addition to decorative arts, glide reflection often occurs in the world of crystals, crystallography considers it to be a separate symmetry operation. Even such simple objects like regular polyhedra can be characterised with a complexity of reflection and rotation symmetry. For example, a cube has several reflection planes and rotation axes. The whole bulk of these simple symmetries makes the group of the symmetry transformations that can be applied to the given regular polyhedron. If one builds a complex structure of a few regular polyhedra, it can lose a few of the symmetries observed in a simple composite polyhedron. However, there may appear new symmetry properties that have not characterised the individual simple building unit. The above series of (simple and complex) geometric symmetry operations could be continued. But these are not the only types of symmetry operations. It is also a symmetry operation if a figure is painted a different colour – if all its other characteristics are left intact. In this instance, the object is a geometric figure, the operation is the change of colour, and the preserved characteristic is everything apart from its colour. Combined symmetries can also be formed from symmetries that have been learned. Examples are the various depictions of perspective in the art (with one or more vanishing points, and aerial symmetry which also produces changes in colour) that combine similitude and affine projection (and colour symmetry) to preserve as many characteristics of the observed world in the picture as possible. The scale of non-geometric symmetries (like charge conjugation in electromagnetism, particle exchangeability, super- and hypersymmetry in field theories, rhythm, functional, i.e., not the morphological, symmetries and asymmetries between the cerebral hemispheres, symmetries in the construction of the genetic code, symmetries in the human perception, symmetries in the statics of architectural structures, including, among others, the quasicrystal and the fullerene structures in stable bindings and their applications in architectural design, …) is very wide. In short, there are simple and complex symmetries, as well as geometric and nongeometric symmetries in all their four combinations. The examples discussed in the first paragraphs were all geometric symmetries. What did the geometric symmetries presented have in common?

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– In each instance, we performed some kind of (geometrical) operation (transformation). – In this process, one or more (geometrical) characteristics of the figure remained unchanged. – This characteristic proved to be invariant under the given transformation (did not change as a result of the operation performed). These three features can be considered as a definition of geometrical symmetries. Let us generalise this in such a way that the interpretation be valid not only for geometrical operations and geometric objects, and not just for geometrical characteristics. In a generalized sense, we can speak of symmetry if – under any kind of (not necessarily geometrical) transformation (operation) – at least one (not necessarily geometrical) characteristic of – the affected (arbitrary and not necessarily geometrical) object remains invariant (intact, unchanged). The generalization, that is, took place regarding three things: – to any transformation, – to an arbitrary object, – to any characteristic. The first generalisation made possible for us not just to look for unchanged characteristics during geometrical operations we have learned (reflection, rotation, translation, etc.). This made possible, for instance, to understand invariance under charge reflection (conjugation) in physics and invariance under the swapping of colours in art. The second generalization makes our concept of symmetry capable of making any kind of object of science or art the subject of a symmetry operation. This paved the way, among other things, for us to use symmetry operations on the abstract objects of physics (like charges). Finally, we allow the constancy of any characteristics to be considered as symmetry (like colour). Of the examples given so far, this was true of electric charge, but any physical quantity considered charge-like can be the object of symmetry, just as can the rhyme of a poem or the motif of a piece of music. This volume is a collection of essays on complex symmetries. Science distinguishes several forms of symmetries. We listed simple symmetries, like reflection, rotation, translation, similitude and a few other simple manifestations of the phenomenon. In the

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instance of a complex symmetry, there is allowed that under two or more transformations, not certainly only one property of an object can be left invariant simultaneously. For example, the well-known helix represents the combination of translational and rotational symmetry. Nature produces more complex symmetries including not only geometric ones. So do the arts. Complex symmetries can be created by the composition of simple geometric and nongeometric symmetries defined in the above way. They can be applied to both simple and complex objects. Simple objects can display complex symmetries (like a frieze), and complex structures can be characterized with only a simple symmetry as well (like our body’s mirror image observed in a looking-glass). The complexity of an object and the complexity of an object’s symmetries can be attributed independently to objects. As against many publications discussing symmetries, this book aims at focusing on the complexity of symmetries (and not that of the objects displaying them). The emphasize is laid not on the phenomena but the analysis of their symmetries. The invited authors were asked to (a) introduce a phenomenon studied by them in short, then (b) present the symmetries of that phenomenon, and (c) analyse the complexity of these symmetries. They were expected to explain the nature of the displayed symmetries, why are they considered symmetries, why are those symmetries complex and how can they be composed of simple symmetries. They attempted to give an insight to the non-specialist readers how individual simple symmetries constitute the complex symmetry of a phenomenon. The scholarly chapters are extended with graphic illustrations by artists. A few of them illustrate complex symmetries in various disciplines. Others explain the complex character of symmetries displayed by a few selected geometric objects. Some of the latter are suitable to represent symmetries observed in the sciences. Both the authors (from thirteen countries) and the topics cover many different disciplines in various sciences and arts, however not all. The theme of the essays is to present how the symmetry of their subject is composed of simple symmetry transformations. The process of the edition included a few iterations between the editor and the authors while the aim of the book was reached. Therefore, the editor expresses his thanks to the authors for their patience during the long-lasted procedure.

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Tamás F. Farkas: BLX, 2003; Paradox structure, illustration to Quantum Scent Dynamics (QSD). [Illustration to Physics]

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Tamás F. Farkas: Paradox Crystal-like Structure X88, 1988; Twisted torus composed of cubes. [Illustration to Crystallography]

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Tamás F. Farkas: Paradox Space Structure VI, 1986; Paradox architectural arrangement. [Illustration to Architecture]

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Tamás F. Farkas: Hypercube Space, 1986; Four-dimensional spatial arrangement at a symmetry angle. [Illustration to Geometry]

Complex Symmetries in Repeating Hyperbolic Patterns Douglas Dunham

Department of Computer Science, University of Minnesota Duluth, 1114 Kirby Drive, Duluth, MN 55812. U.S.A. E-mail: [email protected]; https://www.d.umn.edu/˜ ddunham/.

Abstract: One often uses complex symmetry transformations in order to create art in the hyperbolic plane, producing repeating patterns with corresponding complex symmetries. We explore such transformations for a specific pattern, but these transformations are characteristic of many hyperbolic patterns, including those of M.C. Escher. We also explore the use of color symmetry in creating these patterns, which adds another degree of complexity to the transformations. Keywords: symmetries, hyperbolic, transformations, color symmetry, M.C. Escher, artistic patterns. PACS 2010: 61.50.Ah MSC 2020: 51M10 1 INTRODUCTION There are two aspects of complex symmetries that can be exhibited by repeating hyperbolic patterns. The first deals with combinations of isometries of the hyperbolic plane. The second is the use of color symmetry, which adds another layer of complexity on top of the isometries. A symmetry of a repeating hyperbolic pattern is an isometry of the hyperbolic plane that leaves the pattern invariant, i.e. maps the pattern to itself. The repeating patterns we will consider are composed of copies of a basic subpattern or motif. Thus a symmetry of a pattern will map any copy of the motif to another copy of the motif. Figure 1 shows an example in which the motif is a butterfly. The Dutch artist M.C. Escher mostly used animals for motifs in his repeating patterns.

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Figure 1: A hyperbolic pattern of butterflies.

If the pattern is colored, each motif usually has one color associated to it. In general a symmetry of a pattern will mix up the colors of the motifs, but we will just consider patterns in which this is done in a regular way. These concepts apply not just to hyperbolic patterns, but to patterns in any of the three “classical geometries”: Euclidean, spherical, and hyperbolic. Escher created repeating patterns in each of these geometries, most of them being in the Euclidean plane. In the next section we expand on these ideas, paying particular attention to the symmetry operations. In the following section we examine the invariants of those symmetry operations. In the third section we discuss how complex symmetry operations arise as combinations of simpler symmetries. Finally, we summarize our results. 2 1. THE SYMMETRIES OF REPEATING HYPERBOLIC PATTERNS Initially we consider uncolored repeating patterns of the hyperbolic plane, composed of a theoretically infinite number of copies of a motif. As mentioned above, a symmetry of that pattern is a hyperbolic isometry that leaves the pattern invariant, mapping each copy

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of the motif to another such copy. The set of symmetries of a repeating pattern forms a group, the symmetry group of the pattern. If the pattern is colored, a symmetry of the pattern usually refers to an isometry that keeps the uncolored pattern invariant. In a colored repeating pattern, each motif is assigned a color. So in general a symmetry of the pattern will reassign the color of each copy of the motif. For simplicity, we only consider patterns with a finite number 𝑛 of colors, a characteristic of Escher’s patterns. However, to create an aesthetically pleasing pattern, this should be done in a consistent way. So we first require that a symmetry maps all motifs of one color to a single color, and no two colors go to the same color. Thus each symmetry induces a permutation of the colors. Consequently there is a homomorphism from the symmetry group of a pattern to a subgroup of 𝑆 𝑛 , the symmetric group on 𝑛 colors. In this case we say that the pattern has color symmetry. If that subgroup is transitive, so that all the colors get switched around, we say the pattern has perfect color symmetry. The repeating patterns that we consider all have such color symmetry, as do all of Escher’s colored repeating patterns. Thus we see even by the terms used to discuss (colored) repeating patterns, that topic is intimately tied to symmetry in general. 3 2. INVARIANTS OF SYMMETRIES OF REPEATING HYPERBOLIC PATTERNS Arbitrary isometries of the hyperbolic plane will not leave a repeating hyperbolic pattern invariant, but any isometry that does leave the (uncolored) pattern invariant is called a symmetry of the pattern, as mentioned in the previous section. Some of the symmetries of a colored pattern will permute the colors, and some will fix one color. For example a rotation by 2𝜋/7 fixes the white butterflies of Figure 1 but permutes the other colors. The set of symmetries that fixes one color 𝑖 is a subgroup we call 𝐺 𝑖 of the symmetry group of the pattern. If there are 𝑛 colors, then the intersection from 1 to 𝑛 of all the 𝐺 𝑖 s fixes all the colors. It is a normal subgroup of the symmetry group of the pattern, and its elements leave the entire colored pattern invariant. 4 3. COMPLEX SYMMETRIES BUILT FROM SIMPLE SYMMETRIES Among the “classical” geometries, the hyperbolic plane differs from the sphere and the Euclidean plane in that it cannot be smoothly embedded in the ordinary 3-space in which we live. This was proved by the mathematician David Hilbert over 100 years ago [Hilbert

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Figure 2: Some hyperbolic lines in the Poincaré model.

1901]. Thus we must rely on models of the hyperbolic plane: Euclidean constructs whose parts can be given hyperbolic meanings that are consistent with hyperbolic geometry. One such model is the Poincaré disc model [Greenberg, 2007]. The hyperbolic points of this model are just Euclidean points within a bounding circle. Hyperbolic lines are represented by arcs of circles that are orthogonal to the bounding circle, including diameters as special cases. Figure 2 shows a disc with some hyperbolic lines. For the creation of repeating hyperbolic patterns, this model is preferred by artists since (1) it is conformal, so that the measure of angles is the same as their Euclidean measure and thus motifs retain their approximate shape as they approach the bounding circle, and (2) it is contained in a bounded region of the Euclidean plane so that the viewer can see the entire pattern. Note that it is important that a hyperbolic pattern repeats — i.e. has a theoretically infinite number of motifs — so that we can see its true hyperbolic nature. Otherwise we would just see a finite collection of objects which we would naturally consider to be Euclidean, as in Figure 2 for example. A reflection is the basic isometry in each of the (2-dimensional) classical geometries in that any isometry can be expressed as the composition of at most three reflections. In the Euclidean plane, there are four kinds of isometries: reflection, translation, rotation, and glide reflection. Successive reflections across two lines produce a translation or rotation depending on whether the lines are parallel or intersecting, respectively. If reflection in a

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line is followed by a translation along that line, one obtains a glide reflection, which is thus composed of three reflections. In the sphere there are three kinds of isometries: reflection across a great circle, rotation, and rotatory reflection. Successive reflections across two great circles (which must intersect) produce a rotation about their axis of intersection. Rotatory reflections are analogous to Euclidean glide reflections and can be produced by three successive reflections across great circles. The isometries of the hyperbolic plane are the same as those of the Euclidean plane, with one addition. The additional isometry is produced by successive reflections across two hyperbolic lines that are tangent at a point on the bounding circle. Such an isometry “translates” points along horocycles (represented by circles tangent to the bounding circle) through that point, and is thus called a horolate. Neither Escher nor I have used such an isometry in our repeating hyperbolic patterns, but other artists have. Thus there are four kinds of complex isometries in the hyperbolic plane: translations, reflections, glide reflections, and horolations. In each of the classical geometries, successive reflections in two intersecting lines produce a rotation about the intersection point by twice the angle of intersection of the lines. Thus such a rotation about the intersection point of the two lines to on the right of Figure 2 produces a rotation of 90 degrees about that point since the lines intersect at 45 degrees. A reflection across a line in the Poincaré disc model is given by an inversion in the circular arc representing that line (or an ordinary Euclidean reflection across a diameter in that case). There are no reflections, glide reflections, or horolation symmetries in the pattern of Figure 1. All the symmetries of the uncolored pattern of Figure 1 are either rotations or translations, and thus can be considered to be complex symmetries since they can each be produced by two successive reflections. In fact this is how those complex symmetries were constructed in the computer program that created Figure 1. In order to add color symmetry to a pattern, such as that of Figure 1, one must specify how colors change when a symmetry of the (uncolored) pattern is applied. In the case of perfect color symmetry, this simply a permutation of the colors. In order to do this, we add another level of complexity to the transformation representing the symmetry by encoding the permutation as part of the transformation. Again, this is exactly what was done within the computer program that created Figure 1. All of Escher’s repeating Euclidean patterns, and hyperbolic patterns, as well as most of his spherical patterns may be found in Doris Schattschneider’s book M.C. Escher: Visions of Symmetry [Schattschneider, 2004]. Many of his Euclidean patterns, and two of his four

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hyperbolic patterns possess perfect color symmetry. Other hyperbolic patterns, some with color symmetry may also be found in [Dunham, 1999]. 5 RESULTS We have shown how basic hyperbolic isometries can be composed to form complex symmetries of a hyperbolic pattern. Moveover, we have also shown how the complexity of the symmetries can be increased by adding color symmetry to the symmetry transformations. REFERENCES Dunham, D. (1999), Artistic Patterns in Hyperbolic Geometry, Proceedings of Bridges 1999, pp. 239–250, 1999. Online at: http://archive.bridgesmathart.org/1999/index.html. Greenberg, M. (2007) Non-Euclidean Geometry, 4th ed., New York, W.H. Freeman, Inc., 2007. ISBN 0716724464 Hilbert, D. (1901), Über Flächen von konstanter gausscher Krümmung, of the American Mathematical Society, pp. 87–99, 1901. https://doi.org/10.2307/1986308, https://doi.org/10.2307/2970640 Schattschneider, D, (2004), M.C. Escher: Visions of Symmetry, 2nd ed., New York, Harry N. Abrams, Inc., 2004, 384 pp. ISBN 0-8109-4308-5

IgnoTheory: A Compositional System for Intermedia Art Based on Tiling Patterns and Labelled Graphs Paul Hertz

Independent artist and curator, instructor (retired) in the Department of Film, Video, New Media and Animation and the Department of Art History, Theory and Criticism, School of the Art Institute of Chicago, Chicago, Illinois, U.S.A. E-mail: [email protected], [email protected]; http://paulhertz.net.

Abstract: In this essay, I discuss the rule-based tiling patterns and graphs that I use for algorithmic art and music composition, with particular attention to the symmetries between spatial and temporal concepts of order. The tiling patterns can be regarded as 2D maps which are transformed into graphs with vertices labelled with pitch class names from the Western diatonic musical system. Vertices can also be marked with parameters derived from colouring rules and other combinatorial procedures. Traversal of the graphs can generate material for musical composition and performance. Rotations and reflections of the tiling patterns correspond to transpositions, reorderings and inversions of musical material. Algorithmic operations on visual representations can be mapped onto musical representations and vice versa. Since 1980, the compositional system has been used to generate musical, visual, and theatrical artworks, including an internationally exhibited virtual world (Fools Paradise, 2018). Keywords: tiling patterns, graph theory, visual art, music, algorithmic art, intermedia.

MSC 2010: 00A06 00A08 00A65 00A66 05B45 05C10 52C20

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1 TILING PATTERNS 1.1 The Ignotiles: Rule-Based Tiling Patterns In the fields of design, music, architecture, and media arts a generative system combines elements by following rules to create productions. The productions in these fields are variously functional objects, musical compositions, architectural modules, or visual media. Elements are generally simple structures, the building blocks of the system. Rules govern the combining of elements into productions. Systems typically have two sorts of rules: Functions combine elements together and constraints determine whether a production fits determined criteria. Productions, evidently, are combinations of elements derived from rules. Generative systems resemble mathematical structures, which also define elements, operations and rules for production, and can be formalized and implemented as computer programs. This paper concerns the formal elements of a generative system for interactive multimedia art. The system, which I call IgnoTheory, began with tiling patterns and branched off in many directions (Hertz 1999). I shall concern myself here with the tiling patterns and their derived graphs. The tiling patterns began with a simple modular element, a square hole within a square frame. From 1968 to 1978 I experimented with the geometry of this figure in paintings and drawings and arrived at four unique subdivision tilings, each composed of five shapes: a triangle, two parallelograms, a trapezoid, a “sphinx” shape, and a central rectangle. I assigned letters and names to the tiles, based on a fanciful interpretation of the way the parallelograms are arranged.

Figure 1: The four ignotiles, each composed of a triangle, a trapezoid, two parallelograms, and a “sphinx.”

I called the tiles ignotiles, after a dysfunctional fortuneteller, Ignotus the Mage, who plays a role in the informal, social and symbolic aspects of the system. With a deck of 32 ignotiles, I play the Mage, read the tiles, and see no further than the present.

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Figure 2: The 32 rotations and reflections of the four ignotiles. Each row of four is a 12-point solution. Labelling above columns follows the convention A = butterfly, B = windmill, C = box, D = strider. The 32 tiles became a deck of 32 cards for the dysfunctional fortuneteller Ignotus the Mage.

1.2 Algorithmic Rules for Generating Arrays of Ignotiles 1.2.1 Twelve-point Solutions and Ignoquads Each tile in Figure 1 can be rotated 90 degrees to a new orientation, and each rotated tile can be reflected on a vertical axis to yield 32 oriented tiles (Figure 2). Examination of the diagonal lines connecting the inner square “window” of any ignotile to the outer square “frame” reveals a way to partition the 32 rotations and reflections into eight sets of four different tiles. As shown in Figure 3, diagonal lines in the tiles connect the four corners of the inner square to one of twelve points on the outer square: either to one of its four corners or to one of eight points on its sides. Each tile has three “corner-tocorner” lines (corner diagonals) and two “corner-to-side” lines (side diagonals). One of the corner diagonals lies in between the other two and opposite to an empty corner in the sphinx shape. We’ll call this line the middle diagonal of a tile (shown in red, Figure 3). A 12-point solution is a set of four ignotiles that follows two rules: 1. The set includes a butterfly, windmill, box and strider tile from the 32 tiles. 2. The four middle diagonals and eight side diagonals appear only once in the set of four tiles. The four tiles in a 12-point solution can be arranged in a 2 x 2 square array which I called an ignoquad. The ignoquad became the fundamental structure for developing algorithmic rules within a generative system for fusing visual art and music.

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Figure 3: A 12-point solution set. Left: 12 points on the outer square, corner points in red and side points in green. Center: The 12 diagonal lines that can connect the inner square to the outer square. Right: a 12-point solution arrayed as a 2x2 tile or ignoquad. Each of the four tile types appears in the set, each of the four middle diagonals (in red) appears once, and each of the eight side diagonals (in green) appears once.

1.2.2 Permutations of an Ignoquad There are 4! = 24 permutations of the order of the four tiles in a given ignoquad. To make the permutations more visible, we can shade the tiles along a diagonal through the centre of the tile (Figure 4). The shading reveals 24 unique patterns similar to other shaded square patterns that appear in textiles, ceramics, and architecture over millennia of world cultures (Grünbaum 402–430, Washburn 1988). If we take into account the eight different rotations and reflections of the tiles, there are actually 8 * 24 = 192 different ways to form these patterns from the set of 32 oriented tiles in Figure 3. 1.2.3 Latin Squares A Latin Square is an n x n matrix of n elements where no element is repeated in any row or column (Gardner 1995 ch. 14). We can build a 4 x 4 Latin Square of ignotiles, with some additional restrictions: 1. Select four 12-point solutions from the set of 32 rotations and reflections. 2. Array the 16 selected tiles in a 4 x 4 Latin Square where no tile type (butterfly, windmill, box, strider) is repeated in any row or column, and 3. Each quadrant contains an ignoquad (one of the four 12-point solutions).

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Figure 4: Ignoquad Patterns. Permutations of 4 tiles from a single 12-point solution, shaded to reveal patterns. Notation reads CCW from upper left, where A = Butterfly, B = Windmill, C = Box, and D = Strider.

Figure 5: Left: Patterns for the 12 Latin Squares that fit our rules. The numbers 0, 1, 2, 3 can be indexed to any one of the 24 permutations of the tiles (A, B, C, D). Right: One of the many possible 4 x 4 Latin Square tiling patterns, where 0 corresponds to a Strider Tile, 1 to a Windmill Tile, 2 to a Box Tile and 3 to a Butterfly Tile. The pattern for this ignosquare is found in the lower right corner of the table on the left.

Exactly 12 Latin Square patterns fit these restrictions. The patterns and a tiling derived from one of them are shown in Figure 5. I call a 4 x 4 Latin Square made with four different ignoquads an ignosquare. The total number of ignosquares, counting rotations

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and reflections, is 1680 * 24 * 12 = 483, 840: (8!/4! = 1680) choices from the 8 12-point solutions taken 4 at a time * 24 permutations * 12 Latin Square configurations. I have explored many properties of the ignotiles: colouring problems, algorithms for merging tiles into larger shapes, and computational methods for generating and categorizing ignoquads and ignosquares. I have written about these discoveries elsewhere and created many artworks based on them. In this essay, I will concentrate on the graphs that can be derived from the tiles and their symmetries.

2 GRAPHS 2.1 Musical Composition 2.1.1 Mapping the Ignotiles to the Western Diatonic System The 12-point solution suggested a possible correspondence between the twelve pitch classes of Western diatonic music and the twelve diagonal lines used to draw the tiles. The cycle of pitch classes fit the cycle of diagonal lines around the outer edges of the tiles. There were various ways to map the twelve lines to the pitch classes. A chromatic scale or a circle of fifths were the most obvious. In the end, I used a modified chromatic scale for many experiments, as it yielded interesting tonal material, and followed the order of lines from successive corners of the inner square. In my later collaborations with composer Stephen Dembski, it became apparent that one could arrange the pitch classes in any order that served the purposes of a composition (Dembski 2005). Rhythmic patterns based on the areas of the polygons could be combined with the assignment of pitch classes (for example, three time units for the sphinx, two for the trapezoid, one each for the triangle and the parallelograms, as in Figure 6, below).

Figure 6: Left: Pitch classes are assigned to lines. Here the cycle of pitches forms a modified chromatic scale. Right: Pitch classes are associated with polygons in four tiles that constitute a 12-point solution. A dotted quarter note is used for the sphinx, a quarter note for the trapezoid, and an eighth note for the triangle.

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In the arrangement of pitch classes in Figure 6, triangles and sphinxes are assigned a pitch class from the lines they share with parallelograms. Trapezoids receive the pitches A, C, E♭ and F#, a diminished 7th chord, from the middle diagonals of the tiles. The tonal material from triangles and sphinxes consists of alternating tones and semitones, a series sometimes referred to as the octatonic scale, often employed in jazz improvisation and coincidentally the “Second Mode” of composer Olivier Messiaen (Messiaen 1944). The three pitches present in each tile result in major and minor sevenths with an intervening interval unique to each tile type. 2.1.2 Symmetry under Rotation and Reflection of Tiles

Figure 7: Rotations and reflections of tiles correspond to transpositions and inversions of musical intervals.

When we keep the assignment of pitch classes fixed, rotations of the tiles shift the pitches a minor third up or down (transposition) and reflections of the tiles change the direction of the intervals between pitches (inversion). Transposition and inversion (and also retrograde motion, augmentation and diminution) are transformations found in counterpoint, a musical technique where two or more melodic lines play against one another, and in the composition of European concert music in general. Counterpoint is found in many cultures and flourished in European art music of the late Middle Ages and early Renaissance. Contrapuntal transformation of ordered series (“rows”) of the twelve pitch classes of Western music emerged in the early 20th century in the dodecaphonic or 12-tone method of composition. The ignotiles might be construed as fitting within this tradition, and even more so within its later elaboration as serialism, a music style where not just sequences of tones but durations, timbres, dynamics and other musical elements govern composition. 2.1.3 Parametric Graphs Derived from the Ignotiles In graph theory, a Hamiltonian path is a path that visits each vertex of a graph once (Gardner 1971). A Hamiltonian path on a graph derived from a map would traverse each region of the map only once. A Hamiltonian path or circuit through the shapes in a

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12-point solution can generate a 12-tone row, an ordered sequence of pitches where each pitch is present once only. Polygon shapes can map to sequences of durations. The colours and textures and other parametric values generated from various combinatorial rulesets applied to the tiles can map to sequences of other musical parameters such as instrumentation or dynamics in the derived graphs. If the paths through a tile could be used to generate music by mapping visual properties to musical parameters, they could equally be mapped to actions for performances or any other time-based process. Thus a graph could function as a diagram for a decision-making process. In order to use arrays of ignotiles as elements in decision-making processes, it was necessary to derive graphs from the geometry of the tiles. An ignosquare or any other tiling pattern derived from the 32 ignotiles can be regarded as map. A map can be transformed into a graph by the simple expedient of setting within each “country” a vertex, the “capital city” of the country. If two countries touch, their capitals are connected by a “road,” an edge of the graph. At the end of the process, in accordance with elementary graph theory, you have a graph whose edges and vertices are the dual of the map. Visual attributes of shapes can be mapped onto labels at vertices. The labels can encode parameters of a musical event or any other sort of event we want to create within a decision-making network. I refer to graphs where vertices are labelled with one or more symbolic values as parametric graphs. Initially, the parametric graphs derived this way were not very dense—i.e., vertices were connected by relatively few edges. As a result, they often could not be traversed by a Hamiltonian path, so that “you can’t get there from here” was a real issue in generating a decision-making network. One strategy to increase density was to change the connectivity of tiling patterns embedded in a topological plane, imagining opposite edges to be connected, perhaps with a half-twist through space. In effect, the patterns would be mapped onto 2-D topological surfaces such as a tube, torus, Klein bottle or projective plane, allowing regions on opposite edges to touch one another. Another strategy specified that only the sphinx, triangle, and trapezoid would generate vertices for the graph. The parallelograms would then operate as open connections between the other shapes. On the derived graph the following rule would apply: 1. Two vertices can be connected by an edge if the corresponding shapes can be reached from each other by traversing parallelograms. This made sense musically. The sphinx, trapezoid and triangle mapped to audible durational events, while the parallelograms acted as musical rests, the silences that contain and connect sounds. By combining the two strategies, I could generate graphs

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that were dense enough to allow a variety of paths: not so dense that all vertices were connected to each other, but dense enough to be non-planar. The graph provided small scale structure, at the level of individual musical notes or modes constructed of adjacent pitches. The character of the modes would change as you moved from vertex to vertex. By mapping parameters to larger compositional structures, such as dynamics, tempo, or large scale duration, you could transform a pattern into an entire composition. You could compose scores from musical parameters determined by traversing the graph, or you could even use the graph as a score for improvising musicians. 2.2 Intermedia Art 2.2.1 IgnoTheory: Making Intermedia Art with Parametric Graphs Intermedia is poet Dick Higgins’ term for art that opens a space between different media, that fuses different media, or that uses the compositional techniques of one medium to organize elements of another medium (Higgins 1966). Beginning in 1980 in Spain, I produced various intermedia performances and installations based on the ignotile patterns and graphs. The scores of my intermedia works illustrate the sorts of symmetry transforms that operate within and between the tiling patterns and graphs.

Figure 8: Left: General view of Pond, International Symposium on Electronic Art (ISEA97), Chicago, 1997. Visitors triggered visual and musical events by waving their hands over plastic rods to control animated “agents” traversing a projected image. A modular sculpture representing the ignosquare used as a score covered the floor of the gallery space. Right: A portion of the score for the generation of events.

The interactive multimedia installation Pond used the recorded voices and faces of people who had played Ignotus the Mage’s card game to make ignosquares. One of the ignosquares from the card game and its derived graph provided the control parameters. Software running a graph traversal algorithm controlled musical and visual events. In

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Figure 8, left, a portion of the score shows vertices labelled with a pitch class and a duration derived from the corresponding polygonal shape. An array off to the side of each vertex contains other symbolic values mapped onto the tiling pattern—the first value refers to colour. Edges are labelled with durational values accumulated from the parallelograms they pass through. Note the non-planar character of the graph, which contains a sub-graph of five vertices all connected to each other: a complete 5-graph.

Figure 9: A general view of Fools Paradise and the 48 masks and pavilions that correspond to 48 proverbs from the “Proverbs of Hell” of English poet and artist William Blake.

A virtual world for head-mounted VR, Fools Paradise is based on the “Proverbs of Hell” (1790) of English poet and artist William Blake. The music by composer Stephen Dembski and the virtual architecture I designed were built upon an ignosquare and the parametric graph derived from it (Hertz 2018). Three orthogonal 4x4 Latin Squares encoding colour, texture and a third “symbolic” parameter were mapped onto the tiles and so onto the vertices of the derived graph. The parametric graph is incorporated into the software to generate interaction with the player and produce real-time audio remixes.

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Figure 10: Left: The graph of pitch classes for Fools Paradise and the underlying ignosquare. A Hamiltonian circuit through the ignosquare is marked with a heavy line. Right: A view of the mask at the pavilion for the proverb “Drive you cart and your plow over the bones of the dead.”

3 SYMMETRIES 3.1 Types of Symmetry in the IgnoTheory Generative System Within the IgnoTheory generative system, we evidently have three components to examine for symmetries: two-dimensional tiling patterns and rules for generating them, the graphs derived from the tiling patterns and rules for connecting and traversing them, and the mappings from the tiling patterns to the graphs and vice versa. The rotation and reflection symmetries of ignotiles and ignoquads are summarized in Figures 2 and 4. The ignoquads also have symmetries derived from permutation. The group of all permutations of a set is known as a symmetric group. The ignoquads are a visual representation of a symmetric group of four elements, e.g. {A, B, C, D}. Informally defined, a permutation group is a subset of a symmetric group that remains invariant under a given transformation. The ignoquad permutations can be divided into six permutation groups determined by a cyclic permutation of their elements (Wikipedia contributors 2020). The cyclic permutation annotated (1, 2, 3, 4) sends element 1 to 2, 2 to 3, 3 to 4, and 4 to 1. This is the same as a “right shift” operation on an array. It will produce from the permutation ABCD the permutations DABC, CDAB, and BCDA and then return to ABCD. If we define a function “Cycle” as the application of cyclic permutation (1, 2, 3, 4) to produce a permutation group, then Cycle(ABCD) produces

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permutation group {ABCD, DABC, CDAB, BCDA}. We also define a function “Inverse” which reverses a permutation, so that Inverse(ABCD) = DCBA. Members of Cycle(ABCD), Cycle(ACDB), and Cycle(ADBC) can be paired with their respective inverses in Cycle(ADCB), Cycle(ABDC), and Cycle(ACBD); i.e., given a permutation 𝑥 | 𝑥 ∈ Cycle ADBC → 𝑥 ∈ Cycle Inverse ACBD , and so forth. In the four leftmost columns of Figure 4, we observe that permutations in Cycle(ABCD) and their inverses in Cycle(ADCB), and permutations in Cycle(ACDB) and their inverses in Cycle(ABDC) produce shaded patterns that look like chevrons or stylized arrow shapes with bilateral symmetry on the horizontal or the vertical axis. The remaining permutations, in Cycle(ADBC) and Cycle(ACDB), shown in the two rightmost columns of Figure 4, produce shaded patterns that look like x-shapes or diamond shapes with symmetry on both horizontal and vertical axes. One question to ponder is whether symmetries of this sort, which are a result of the geometry indexed by the permutation groups and not of the permutation groups themselves, have some counterpart in the derived graphs. Of course, a mapping should work the other way around, so that operations on properties in the graphs obtain the change of a property in the tiling patterns, too. 3.2 Symmetries between Maps and Graphs The procedure that produces parametric graphs from the ignotiles is a modified version of the classic derivation of graphs from normal maps. In the classic derivation, the connectivity of regions of the map is entirely described by its dual graph. Other information, including the geometry of the maps, is of course not encoded in the graph. Ignotiles are not normal maps, and the procedure for producing a graph does not capture the precise connectivity of the tiles; however, with sufficient contextual information, one can go back and forth between the ignotiles and derived graphs. For our purposes, we simply assume that we know both the array of tiles and its derived graph. We have already uncovered one potential symmetry mapping between tiling patterns and graphs: rotations and reflections of the tiles correspond to transpositions and inversions of the sequences of pitches in the graphs (Figure 7). This relation holds because of the nature of the mapping between the tiling patterns and their derived graphs: an ordered sequence of points around a square acts as an index to an array such that each point indexes a discrete value. The values in the array may be points or pitch classes or numbers, but the indexing schema remains unchanged. A tile can be represented as an index into the array plus a series of offsets from the index that access

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the values associated with endpoints of lines within the tile. Rotation of the tile corresponds to shifting the index into the array and reading new values at specified offsets, wrapping around to the beginning of the array if the index exceeds the number of elements in the array. For 90° rotations, the index is shifted by 3 places. Reflecting the tiles both shifts the pointer and flips the sign of the offsets. The individual tiles, A, B, C and D, or butterfly, windmill, box and strider, map onto triangles in the derived graph. Each triangle supplies three sequential tones to the 12tone row associated with a Hamiltonian path through an ignoquad. The graphs in Figures 8 and 10 show how the triangles fit the ignotiles; Figure 10 also includes a Hamiltonian circuit over four ignoquads. If we rotate or reflect an ignoquad, we transpose or invert the row. If we only rotate or reflect the individual tiles in an ignoquad, we transpose or invert and also reorder the row by changing the connectivity of the graph and the Hamiltonian path. Certain orderings of pitch classes reveal symmetries within the indexing schema as it applies to the ignotiles.

Figure 11: Correspondences between the indexing schema and ordered pitch class values. Pitches are listed as pitch class names (Af = A flat, Cs = C sharp). Lines and tiles associated with indices and pitches are flagged with colour. The “group” is either an All Interval (AI) Group (1, 2) or a diminished chord (3).

In music theory, an All Interval or AI group contains four pitches whose distances from one another comprise all the possible intervals in the chromatic scale. There are two AI

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groups, whose intervals are either 1-3-2 or 1-2-4. Although these groups apply to music theory, they belong to group theory for modulo-12 “clock” arithmetic, whose application to 12-tone Western musical theory is well-documented (O’Connell 1968). Looking at array indices and the chromatic scale, we can determine that the values at the corner diagonals of a 12-point set of tiles (top row of figure 11) form a 3-3-3 interval group (0 3 6 9 and A C Ef Fs). The side diagonals of tiles A and C make a 4-2-1 AI group (10 2 4 5 and G B Cs D); the side diagonals of tiles B and D make a 1-3-2 AI group (7 8 11 1 and E F Af Bf). The AI group and 3-3-3 group properties are inherent in the indexing schema and could be considered a product of the original rule for the ignotiles: arrange a triangle, two parallelograms, a trapezoid, and a sphinx shape around a square with a square hole in its centre. The chromatic scale is an ordered sequence of twelve values, which can simply be the values of the indices—so of course, the chromatic scale exhibits the same properties. Using a circle-of-fifths instead provides a 2-3-1 group (F G Bf B) and a 1-2-4 group (Cs D E Af). Modified chromatic and circleof-fifths mappings where we swap the values at side diagonals (indices 1, 2, 4, 5, 7, 8, 10, 11) also conserve the AI/3-3-3 property. As far as I know, no other mappings of pitch classes to indices will conserve this property. The AI/3-3-3 property could be construed as a counterpart in the derived graphs to the symmetries of the shaded shapes in the ignoquads, which are associated with permutation subgroups (see section 3.1). In the derived graphs, each tile is represented by a triangle of vertices. Triangles from adjacent tiles will necessarily have paths connecting them. Triangles from diagonally opposite tiles may be connected, but only if the derivation rule permits. In the ignoquads mapped to the Cycle(ACDB), Cycle(ABDC) and Cycle(ADBC), Cycle(ACDB) permutation groups, A and C are always adjacent and B and D are always adjacent. Their members are assured of producing graphs that contain AI groups. The Cycle(ABCD) permutation and its inverse Cycle(ADCB) are unlikely to generate members with AI groups, though there will be other tonal material possible in each permutation group’s derived graphs that are unique to the group. Though not aesthetically dramatic, as the shaded tiling patterns certainly are, the availability of AI groups in a derived graph is clearly also a property of the permutation group for the ignoquads.

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3.3 Mapping Arbitrary Values from Graphs to Maps

Figure 12: The BCDA ignotile and its derived graph, which includes a complete graph of six vertices. The tile illustrates another way of representing the derivation rule, where adjacent parallelograms are merged and represented by a single “hub” vertex. All other vertices connected to the hub are connected in the graph.

This lack of aesthetic drama returns me to the awareness that music is not math, and that graph theory, modular arithmetic, and array transformations, though marvellous in their capacity to order our thinking and expand our panoply of forms, do not capture music or visual pattern making, which arise from sensory and cultural experiences. I would like to close by discussing transforms from the graphs back to the tiling patterns/maps that do not conserve symmetries arising from sequentially ordered pitch classes (such as the AI/3-3-3 symmetry) but that may prove useful for intermedia composition. The vertices of the parametric graphs can be labelled with parameters that map to visual attributes of the polygons in the tiling patterns such as colour, texture, and shape. They may also be labelled with values that are simply the result of combinatorial processes without a corresponding visual attribute. They may be labelled with values that correspond to the 12 lines that link the interior and exterior squares of a tile. I have chosen to put the lines into one-to-one correspondence with musical pitch classes and encountered some evidence that visual and musical art may be composed using this correspondence as a source of intermedia structure. The ordering of the pitch classes, which at first adhered to cyclic patterns such as the chromatic scale or circle-of-fifths, is ultimately arbitrary. To proceed further musically, it would make sense to regard the connectivity of the graphs as a practical element for intermedia composition independent of the pitch classes. For example, many complete graphs are generated as a result of the derivation rule. Figure 12 shows how a complete 6-graph can be derived from the BCDA ignotile. This and similar subgraphs can be populated with the pitch classes of hexatonic or pentatonic scales or whatever material a composer finds apt for a

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particular composition. It is no longer necessary to derive the sequencing of pitches (i.e., melody) from the combinatorial math. The decision-making process may use any other parametric values as desired, but at least one central aspect of the IgnoTheory system can be freed up—by another rule perhaps! That would be: 1. The order of the array of twelve pitch classes for each ignoquad may be determined by the composer as her craft and experience require. This is not a mathematical rule, I know, but it may open the way for new intermedia art to emerge. The pitch classes will still map back to the tiling pattern and can operate in events at precise locations in a virtual world. Indeed, VR seems to be an optimal space for these theoretical musings to be transformed into experiences. It will probably not surprise you if I say that I am working on a VR world designed around this new un-ruly rule. And I hope to see you there, dear reader.

REFERENCES Dembski, S. (2005) An Idea of Order, Perspectives of New Music, 43, 2/44, 1 (summer 2005/ winter 2006): 403-424. Gardner, M. (1966) New Mathematical Diversions, Spectrum Series, Revised edition (CD-ROM) 2005, Washington, DC: Mathematical Association of America. Ch. 14 (162–172) “Euler’s Spoilers: The Discovery of an Order-10 Graeco-Latin Square”. Gardner, M. (1971) Martin Gardner’s 6th Book of Mathematical Diversions from Scientific American, Revised edition (CD-ROM) 2005, Washington, DC: Mathematical Association of America. Ch. 10 (91–103) “Graph Theory”. Grünbaum, B., and Shephard, G.C. (1986) Tilings and Patterns, New York: W.H. Freeman. Hertz, P., Ox, J., Blom, I., Polli, A., and Spielmann, Y. (1999) Erasing Boundaries: Intermedia Art in the Digital Age, Panel description, ACM SIGGRAPH Conference 1999, Los Angeles. Retrieved 17:26, November 12, 2020, from http://digitalartarchive.siggraph.org/wpcontent/uploads/2018/01/hertz_panel.pdf. Presentation notes: retrieved 17:26, November 12, 2020, from http://paulhertz.net/worksonpaper/ignotheo.pdf Hertz, P., and Dembski, S. (2018) Fools Paradise: Intermedia in VR: Madrid, Spain, xCoAx 2018: Proceedings of the Sixth Conference on Computation, Communication, Aesthetics and X. Retrieved 17:26, November 12, 2020, from http://2018.xcoax.org/pdf/xCoAx2018-Hertz.pdf Higgins, D. (1966) Intermedia, New York City: Something Else Press, revised 1981. Reprinted in Leonardo, 34, 1 (2001): 49-54. The MIT Press. https://doi.org/10.1162/002409401300052514 Messiaen, O. (1944) The Technique of My Musical Language, Paris: Éditions Musicales Alphonse Leduc et Cie. English edition translated 1956, John Satterfield. Retrieved 17:26, November 12, 2020, from https://monoskop.org/File:Messiaen_Olivier_The_Technique_of_My_Musical_Language.pdf O'Connell, W. (1968) Tone Spaces: Die Reihe, no. 8: Theodore Presser Company of Bryn Mawr, Pennsylvania, in association with the Universal-Edition. Retrieved 17:26, November 12, 2020, from https://monoskop.org/images/e/ec/Die_Reihe_1-8_EN_1957-1968.pdf#page=398 Washburn, D.K., and Crowe, D.W. (1988) Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Seattle, WA: University of Washington Press. For examples of patterns from various world

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cultures composed of squares or rectangles shaded along a corner-to-corner diagonal, see pp. 102, 123, 173, 187, 195, 215, 265. Wikipedia contributors (2020, September 23) Permutation group, in Wikipedia, The Free Encyclopedia. Retrieved 17:26, November 12, 2020, from https://en.wikipedia.org/w/index.php?title=Permutation_group&oldid=979963440

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Patrice Jeener: Gravure, tetrahedral; regular polytope including 120 cells with tetrahedral symmetries.

Symmetries of Maps on Surfaces Ashish Kumar Upadhyay

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi, 221005. INDIA E-mail: [email protected]; . ORCID: 0000-0001-6307-6799

Abstract: This is a brief overview of symmetry, its complexity and its applications of maps on surfaces. The details of this can be found in the references given in the article. Keywords: Combinatorial Topology, Combinatorial Manifolds, Polyhedral Manifolds, Automorphism groups. MSC 2010: 52B70, 57M20, 57M10 1 DEFINITIONS AND PRELIMINARIES In this article, a surface Σ will mean a connected, compact and closed 2-dimensional manifold. An embedding of a finite simple graph Γ on the surface Σ is called a map (see Jones et al., 1978), 𝑀, on Σ if the closure of each component of Σ \ Γ is a 2-disc with polygonal boundary, 𝑓 , and such that non-empty intersection of any of the two such discs is either an edge, 𝑒, or a vertex, 𝑣, of Γ. We say that 𝑣, 𝑒 and 𝑓 are, respectively, vertices, edges and faces of the map 𝑀. If each 𝑓 in 𝑀 has exactly 3 sides then 𝑀 is called a triangulation of Σ. On the other hand, if each face of 𝑀 is a 𝑝-gon and has 𝑞 many of them incident at each vertex then 𝑀 is said to be {𝑝, 𝑞} equivelar or is of type 𝑝 𝑞 , see 𝑞2 𝑞𝑘 Brehm et al., 2008 Similarly, if 𝑀 is of the type 𝑝𝑞1 1 𝑝 2 . . . 𝑝 𝑘 then we say that 𝑀 is 𝑞1 𝑞2 𝑞𝑘 {𝑝 1 𝑝 2 . . . 𝑝 𝑘 } semi-equivelar map. The idea underlying study of equivelar maps is the extension of maps which lie on the surface of Platonic solids and the idea underlying study of semi-equivelar maps is the extension of maps on surface of Archimedian solids, see Grunbaum et al., 1987, Chavey (1989), Macmullen et al., 1982, Macmullen et al., 1983

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Darvas (ed.), Complex Symmetries, https://doi.org/10.1007/978-3-030-88059-0_4

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Corresponding to a map 𝑀 we can form a set 𝐾 (see Vince, 1983) which consists of all the vertices, edges and faces of 𝑀 as its elements. We then call 𝐾 a polyhedral complex associated with 𝑀. It is easy to see that 𝐾 is a combinatorial version of 𝑀 and the terminologies of 𝑀 are inherited verbatim for 𝐾 and we will not differentiate between 𝑀 and 𝐾 as far as terminology or concepts are concerned in this article. We denote the set of vertices, edges, and faces in 𝐾 by 𝑉 (𝐾), 𝐸𝐺 (𝐾), and 𝐹 (𝐾), respectively. See Altshuler et al., 1992, Altshuler et al., 1996, Ellingham et al., 2005. Two polyhedral complexes 𝐾1 and 𝐾2 are said to be isomorphic if there exists a bijection 𝑓 : 𝑉 (𝐾1 ) → 𝑉 (𝐾2 ), which extends linearly to each 𝜎 ∈ 𝐾1 and such that 𝜎 ∈ 𝐾1 if and only if 𝑓 (𝜎) ∈ 𝐾2. We write 𝐾1  𝐾2 if 𝐾1 is isomorphic to 𝐾2. See Brehm et al., 1997, Brehm 1990, Brehm et al., 2002.

An isomorphism from a polyhedral complex 𝐾 to itself is called an automorphism. The set of all automorphisms of 𝐾 is denoted by 𝐴𝑢𝑡 (𝐾). Naturally, 𝐴𝑢𝑡 (𝐾) forms a group under the operation of composition of maps, and is called the group of automorphisms of 𝐾. A connected polyhedral 2-manifold 𝑋 is said to be vertex-transitive map if 𝐴𝑢𝑡(𝑋) acts transitively on 𝑉 (𝑋) see Datta 2005, Datta et al., 2001, Datta et al., 2005, Lutz et al., 2010. If the order of an element of automorphism of a polyhedral complex is two, then it is called an involution. If 𝐾 is invariant under a group action, where the group is generated by involution of the vertex set of 𝐾 which fixes no faces, then the manifold is called centrally symmetric. Central symmetry does have broad applications in mathematics Lutz, 1999, Maity et al., 2019. 2 THE SYMMETRY CONSIDERATIONS Symmetry is a phenomenon which occurs naturally in the form of processes or structures which often appear abundantly in nature. The underlying idea is to understand the subtle concepts of minimizing energy and exhibiting beauty. But then what is beauty will be a question remaining to be answered. In the objects which we are considering the main purpose is to classify them and also to understand the features and attributes of underlying objects which remain invariant under symmetrical transformations. We also try to understand their underlying beauty and seek to understand their application to other domain of knowledge. As a consequence we sometimes stumble upon important results which arise as an application of our investigation. Thus we attempt to construct, classify, and recognize such objects. Symmetry considerations play a key role, both for theoretical work and for practical computations. In what follows we will justify that some softwares are used to

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perform these activities leading to new results about objects through computer search. Availability of fine computational facility may greatly increase the range of what can be calculated manually. The use of computer to discover new objects gives an idea about the general situation and significatly aids in formulating and proving general results. Thus proving existence of objects with a predetermined symmetry becomes an economical exercise whereas at times exhaustive searching by computers is also undertaken. See Conder (2009), Kohler et al., 2005, Lutz et al., 2010, Sulanke, et al., 2009 A key element in analysing the symmetries lies in understanding the action of the automorphism group of the object. What are all the various ways in which groups can act on combinatorial objects? Applications of theory of maps on surfaces range from combinatorics to geometry, number theory, and algebra and for instance, in information theory, computer science, and physics and chemistry. See Cromwell 1997, Coxeter et al., 1980, Debashis et al.,2020 The study of maps on surfaces is indeed related to the study of theory of groups Brahana 1927, Coxeter et al., 1980. From Jones et al., 1978 we can see that maps on surfaces can also be alternatively considered as subgroups of certain groups. These groups in turn also lead to determination of Automorphism group of such maps. There has been some work in this direction to determine as to which groups can occur as automorphism of surfaces of a given genus. As an interesting outcome of this fact we can use algebraic techniques to construct maps on these surfaces and study their symmetries. Few computational packages involving Algebra or group such as MAGMA and GAP has hence been used to determine such maps using group theoretic techniques as well (see, e.g. Conder (2009),Kohler et al., 2005, Karabas et al., 2007, Karabas et al., 2012, Lutz, 1999, Lutz et al., 2010 etc.). 3 COMPLEX SYMMETRIES In the cases we consider in this article the features and attributes which remain invariant under the action of the automorphism groups are all the topological properties together with many combinatorial properties as well. It is natural to study the objects under consideration by decomposing them into smaller familiar pieces. To this author’s understanding the constraints on number of such pieces seems to ensure, in a subtle way, the minimization of energy where as the invariance under permissible transformations (automorphisms) preserves beauty. In addition to classify the maps obtained, we have defined a graph 𝐺𝑖 𝑠 (see Datta et al., 2001, Datta et al., 2005, Datta et al., 2006 to study the combinatorial equivalence of

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these objects. However if the objects are isomorphic the graphs must be isomorphic. On the other hand the isomorphic graphs are also surprisingly useful in constructing isomorphisms of the objects. It goes without saying that if graphs are non-isomorphic the objects can not be isomorphic too. Additionally, the combinatorial features such as the cliques, the characteristic polynomials of the incidence matrices of 1-skeleton of these maps etc also remain invariant as do the number of various faces in each dimension. It is well known that the rigid motions in dimensions 2 are generated by translations and rotations. The individual symmetries in the case of maps on surfaces can be taken as those derived from these individual symmetries by combining them using composition. The maps on surfaces arising due to the embeddings of graphs which constitute the maps lift to the universal cover of the surfaces used to construct the maps and forms a tesselletation there. We are already aware of isometry groups of simply connected 2 dimensional manifolds and the resulting map on the surface is a consequence of going to quotient of these isometry groups by their discrete subgroups. These quotient groups then form automorphism group of the map on surfaces (see Coxeter et al., 1980). The automorphism group naturally acts on the maps via automorphisms. The size of the automorphism group may be considered, intuitively, as a measure of how symmetrical the object under consideration is. There are other aspects of the action of automorphism group on 𝑀. If the action of 𝐴𝑢𝑡 (𝑀) on 𝑀 is transitive on 𝑉 (𝑀) or 𝐸 (𝑀) we say that 𝑀 is a vertex transitive map or edge transitive map. Similarly if 𝐴𝑢𝑡 (𝑀) acts transitively on the set 𝐹 of face vectors (𝑣, 𝑒, 𝑓 ) (also called a flag), comprising of ordered triples of vertices, edges and faces such that the vertex is contained in the edge and edge in face then the map 𝑀 is said to be flag transitive or regular. Regular maps are the most symmetrical objects in this field. A significant amount of work has been made to study the vertex transitive maps, see Babai 1991,Datta et al., 2006,Karabas et al., 2007,Karabas et al., 2012, Lutz, 2010, Lutz, 1999, Lutz et al., 2010, Pellicer, 2014, edge transitive maps Karabas, 2020, Orbanic et al., 2011, Siran et al., 2001, Wilson, 1997 and regular maps Brahana 1927,Conder (2009),Conder et al., 2001,Conder et al., 1995,Sherk, 1959,Vince, 1983. 4 SOME RESULTS Using the analysis mentioned in previous sections it was natural to consider studying such objects. We briefly state some important results which we have been able to obtain over last few years.

Symmetries of Maps on Surfaces

39

Apart from some results on torus and Klein bottle Datta et al., 2005, we have been able to show that (see Datta et al., 2006) There are exactly six equivelar triangulations on double torus up to isomorphism. Three of these are vertex transitive and remaining are not. As an important corollary to the above result we obtain (see Datta et al., 2006) There are exactly twelve equivelar maps on double torus. It is widely known that equivelar maps do not exist on the surface of Euler characteristics - 1, this necessitated study of semi equivelar maps on this surface. We initiated this study in Upadhyay et al., 2012 and showed that (see Debashis et al.,2020 , Tiwari et al., 2017) there are at least 17 non-isomorphic semi -equivelar maps on the surface of Euler characteristic -1. None of these maps are vertex transitive. We have also been successful in determine semi-equivelar maps and their symmetry groups on Torus, Klein bottle and surfaces of Euler characteristics -2 (see Debashis et al., 2020,Debashis et al.,2020 ,Debashis et al.,2020,Tiwari et al., 2017). As a consequence of above results we were able to answer questions posed by Negami and Nakamoto Negami et al., 2001 in Lutz et al., 2010 and were able to add some results towards answering Grunbaum’s Conjecture Maity et al, 2020. Acknowledgements The author would like to express his gratitude to Spyros Magliveras for his invitation and support throughout the preparation of this paper. Thank is also due to the anonymous referee whose suggestions lead to improvement in the presentation of this article. The author would like to thank SERB DST for its grant no MTR/2020/000006 under MATRICS scheme. REFERENCES Altshuler, A. and Brehm, U.(1992) Neighborly maps with few vertices, Discrete Math. 8, 93–104. Altshuler A., Bokowski, J., Schuchert, P.(1996) Neighborly 2-manifolds with 12 vertices, J. Comb. Theory, 75, 148-162. Babai, L.(1991) Vertex transitive graphs and vertex transitive maps, J. Graph Th. 15, 6, 587-627. Brahana, H. R. (1927) Regular maps and their groups, Amer. J. Math., 49, 268-284. Brehm, U., Schulte, E. (1997) Polyhedral maps, In : Goodman J. E. and Rourke J. O’., eds. Handbook of discrete and computational geometry , Second edition, CRC Press , 477 - 488 Brehm, U.(1990) Polyhedral maps with few edges, In: Bodendiek R. and Henn R., eds. Topics in Combinatorics and Graph Theory: Essays in Honour of Gerhard Ringel, Physica-Verlag Heidelberg, 153-162. Brehm, U., Datta, B. and Nilakantan, N.(2002) The edge-minimal polyhedral maps of Euler characteristic −8, Beitr. Algebra. Geom., 43, 583-596.

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Brehm, U. and K¥uhnel, W.(2008) Equivelar maps on torus, Europ. J. Comb., 29, 1843-1861. Cromwell, P. R. (1997) Polyhedra, Cambridge University Press: New York, 1997. Chavey, D. (1989) Tilings by regular polygons-II, Computers Math App., 17, 147-165. Conder, M. D. E. (2009) Regular maps and hyper maps of chi -1 to -200, J. Comb. Theory, 99, 455-459. Conder, M. D. E. and Dobcs´anyi, P.(2001) Determination of all regular maps of small genus, Journal of Comb. Theory, 81, 224-242. Conder, M. D. E. and Everitt B. (1995) Regular maps on non-orientable surfaces, Geom. Dedicata, 56, 209-219. Coxeter, H. S. M. and Moser, W. O. J. (1980) Generators and Relations for Discrete Groups; Springer-Verlag: Berlin, 1980. Datta, B. (2005) A note on the existence of {𝑘, 𝑘 }-equivelar polyhedral maps, Beitr 𝑎ge ¥ zur Algebra und Geometrie, 46, 537 - 544. Datta, B. and Nilakantan, N. (2001) Equivelar Polyhedra with few vertices, Discrete Comput. Geom., 26, 429-461. Datta, B. and Upadhyay, A. K.(2005) Degree-regular triangulations of torus and Klein bottle, Proc. Indian Acad. Sc.(Math. Sc.), 115, 279-307. Datta, B. and Upadhyay, A. K. (2006) Degree-regular triangulations of the double torus, Forum Math., 18, 1011-1025. Bhowmik D. and Upadhyay A. K.(2020) Classification of semi-equivelar maps on the surface of Euler characteristic −1, Indian J. Pure and App. Math., (to appear) Bhowmik D., Maity D., Upadhyay A. K. and Yadav B. P. (2020) Semi-equivelar maps on the surface of Euler genus 3, https://arxiv.org/abs/2002.06367, preprint. Bhowmik D. and Upadhyay A. K. (2020) A classification of semi-equivelar maps on surface of Euler characteristics −2, https://arxiv.org/abs/1904.07696, preprint. Bhowmik D., Maity D., Yadav B. P. and Upadhyay A. K., (2020) New classes of quantum codes associated with maps on surfaces, preprint. Edmonds, A. L., Ewing J. H. and Kulkarni, R. S.(1982) Regular tessellations of surfaces and (p, q, 2)-triangle groups, Annals of Math., 116, 113-132. Ellingham, M. N. and Stephens, C. (2005) Triangular embeddings of complete graph (neighborly maps) with 12 and 13 vertices, J. Combi. Designs, 13, 336-344. Gr¥unbaum, B. and Shephard G. C. (1987) Tilings and Patterns, W. H. Freeman and com., New York, 1987, pp. 57 - 111. Jones, G. A. and Singerman, D. (1978) Theory of maps on orientable surfaces, Proc. London Math. Soc. , 37, 273-307. K¥ohler, E. G. and Lutz, F. H. (2005) Triangulated manifolds with few vertices, arXiv:math.GT/0506520, 2005. Karabab, J. (2020) https://www.savbb.sk/ karabas/edgetran.html Karabas, J. and Nedela, R. (2007) Archimedean solids of genus two, Electronic Notes in Discrete Math., 28, 331-339. Karabas, J. and Nedela, R. (2012) Archimedean solids of higher genera, Mathematics of Comput., 81, 569-583. Lutz, F. H.(2010) Equivelar and d-covered triangulations of surfaces. II. Cyclic triangulations and tessellations, arXiv:1001.2779, preprint.

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Lutz, F. H.(1999) Triangulated manifolds with few vertices and vertex-transitive group actions, Shaker Verlag: Aachen, 1999. Lutz, F., Sulanke, T., Tiwari, A. K. and Upadhyay, A. K. (2010) Equivelar and d-covered triangulations on surfaces - I, arXiv:1001.2777, preprint McMullen, P., Schulz, Ch. and Wills, J. M.(1982) Equivelar polyhedral manifolds in 𝐸 3 , Isr. J. Math. 41, 331-346. McMullen, P., Schulz, Ch. and Wills, J. M.(1983) Polyhedral 2-manifolds in 𝐸 3 with unusually lerge genus, Isr. J. Math. , 46, 127-144. Maity D. and Upadhyay A. K. (2019) On centrally symmetric manifolds, J. Ramanujan Math. Soc., 34, 21–27. Maity D. and Upadhyay A. K. (2020) Hamiltonicity of a class of toroidal graph, Math. Slovaca, 70, 497–503. Negami, S. and Nakamoto, A.(2001) Triangulations on Closed Surfaces Covered by Vertices of Given Degree, Graphs and Combinatorics, 17, 529 - 537. Orbani´c A., Pellicer D., Pisanski T. and Tucker T. W.(2001) Edge-transitive maps of low genus, Ars Math. Contemp., 4, 385–402. Pellicer D., (2014) Vertex-transitive maps with Schläfli type {3, 7}, Discrete Math., 317, 53-74. Sherk, F. A. (1959) The regular maps on a surface of genus three, Canad. J. Math., 11, 452-480. ˇSir´aˇn J., Tucker T. W. and Watkins M. E.(2001) Realizing finite edge-transitive orientable maps, J. Graph Theory, 37, 1–34. Sulanke, T. and Lutz, F. H.(2009) Isomorphism free lexicographic enumeration of triangulated surfaces and 3-manifolds, Eur. J. Comb., 30, 1965-1979. Tiwari, A. K. and Upadhyay, A. K. (2017) Semi-equivelar maps on the torus and the Klein bottle with few vertices, Math. Slovaca, 67, 519–532. Tiwari A. K. and Upadhyay A. K. (2017) Semi-equivelar maps on the surface of Euler characteristic -1, Note di Mat., 37, 91–102. Upadhyay, A. K., Tiwari, A. K. and Maity, D. (2012) Semi-equivelar maps, Beitr 𝑎ge ¥ zur Algebra und Geometrie, 55, 229–242. Vince, A. (1983) Combinatorial Maps, J. Comb. Th. Ser B , 34, 1-21. Vince, A. (1983) Regular Combinatorial Maps, J. Comb. Th. Ser B, 35, 256–277. Wilson, S. (1997) Edge-transitive maps and non-orientable surfaces, Math. Slovaca, 47, 65–83.

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P. JEENER

Patrice Jeener: Gravure, dodecahedral; regular polytope including 120 cells with dodecahedral symmetries.

Symmetry in Projection of 4-Dimensional Regular Polychora Koji Miyazaki1* and Motonaga Ishii2

1

Graphic Scientist, Architect, D.Eng., Professor Emeritus of Kyoto University, Japan

E-mail: [email protected] * Corresponding author 2

Computer Scientist, Physicist, PhD, Engineer of BANDAI NAMCO Research Inc., Japan

E-mail: [email protected]

Abstract: The ultimate end of this paper is to show the practical usefulness of symmetry seen in a projection of 4-dimensional polychora into 3- or 2-space. The main tools used for this purpose are familiar rotational and reflectional symmetry relating to the simple orthogonal and one-point perspective projections of regular polychora. As a result, some unique aperiodic planar patterns and spherical structures are derived. Keywords: 4-dimensional space, Projection, Symmetry, Regular Polychora

1 INTRODUCTION In order to understand and represent 4-dimensional shapes, we must rely on various projections into 3- or 2-dimensional space. Arbitrary projections of random figures are of no practical use because of their confusing appearance. On the contrary, well-regulated orthogonal parallel and one-point perspective projections of regular polychora are useful because they show understandable rotational and reflectional symmetry features which allow for minute examination and representation of 4-dimensional shapes. These in turn can serve further utilitarian or decorative purposes.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Darvas (ed.), Complex Symmetries, https://doi.org/10.1007/978-3-030-88059-0_5

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K. MIYAZAKI, M. ISHII

2 SYMMETRY Figure 1 shows the outline of rotational and reflectional symmetry, each of which is known as direct and indirect symmetry. On a 2D-plane, there is rotational point symmetry around a 0D-point and reflectional line symmetry in a 1D-line. Of them, rotational symmetry is derived from double reflection symmetries. In 3-space, there are symmetries by point inversion, rotation around a line, and reflection in a plane. Inversion symmetry is derived from 3 times reflection symmetries or from a combination of rotation and reflection symmetries (Cromwell, 1997, Darvas, 2007). In 4-space, there are symmetries by inversion in a point, inversion in a line, rotation around a plane, and reflection in a hyperplane. 4-dimensional point inversion is derived from 4 times reflection symmetries or from a combination of rotational and reflectional symmetries. That is, every symmetry is governed by rotational or reflectional symmetry. So far as regular figures are concerned, rotational symmetry is classified into regular polygonal, polyhedral, or polychoral group and reflectional symmetry into Coxeter group.

Figure 1: Outline of symmetry on a 2D-plane (upper left half), in 3D-space (upper right half) and in 4D-space (lower)

3 PROJECTION There are two kinds of well-known projections: parallel projection whose projecting lines are mutually parallel - similar to Sun rays during daytime - and perspective projection whose projecting lines emerge radially from a projecting point - similar to the rays emitted by a lamp during nighttime. A particular case of parallel projection is called orthogonal projection, where the projecting lines intersect orthogonally the image’s plane, whereas one-point perspective - which has only one vanishing point - is one of the most commonly used forms of perspective projections.

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Figure 2 shows their outlines by using orthogonal coordinate axes. In the case of orthogonal projection in 3-space, the projecting lines run parallel to the Z-axis and the resulting projection is drawn onto the XY-plane. Similarly, in 4-space, projecting lines run parallel to the U-axis and the resulting projection is put in XYZ-space. A projection onto the XY-plane is sometimes called a plan, whereas the XZ projection an elevation and the XU projection a hyper-elevation. On the other hand, in the case of the one-point perspective in 3-space, a projecting point S (0,0,s) is set on the Z-axis and the resulting projection is drawn onto the XY-plane, where x’=sx/(s-z), y’=sy/(s-z). Similarly, in 4space, a projecting point S (0,0,0,s) is set on the U-axis and the resulting projection is put in XYZ-space, where x’=sx/(s-u), y’=sy/(s-u), z’=sz/(s-u) in 4-space.

Figure 2: Outline of orthogonal projection (left) and one-point perspective (right) depicted by using orthogonal coordinate axes.

Figure 3 shows examples of the orthogonal projections of a 4-dimensional orthogonal coordinate system of axes. In the case of 3-projection, they can be embedded into a regular tetrahedron, and in case of 2-projection, into a plan-elevation pair with a special sideview of a regular dodecahedron. In this case, on XY-plane as a plan, OX(=OY):OZ (=OU) =1:τ and ∠XOY (=36°)✕3=∠XOU=∠YOZ=∠ZOU, on XZ-plane as an elevation, the true length in 4-space can be taken on the U’-axis, and the 3-dimensional isometric axes O”-X”Y”Z” appear in the special side-view which is derived from projection along U’axis, where ∠XOY=∠YOZ=∠ZOX=120° and O” X”= O”Y” =O”Z” (Miyazaki, 2005, 2020).

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Figure 3: Symmetric orthogonal 3-projection (left) and 2-projections (right) of 4-dimensional orthogonal coordinate axes.

4 REGULAR POLYCHORA Each regular polychoron, which is the 4-dimensional analogue of a 3-dimensional regular polyhedron, has only one kind of regular n-gonal face, denoted by the Schläfli symbol {n}. When a fixed number m copies of an {n} fits together around every vertex, a regular polyhedron {n,m} is constructed, and a fixed number l copies of an {n,m} fits together around every edge, a regular polychoron {n,m,l} is obtained. There are 6 kinds of regular polychora: The 5-cell {3,3,3} as a 4-dimensional regular tetrahedron, the 8-cell {4,3,3} as a 4-dimensional cube, the 16-cell {3,3,4} as a 4-dimensional regular octahedron, the 24-cell {3,4,3} as a nearly 4-dimensional cuboctahedron or rhombic dodecahedron, the 120-cell {5,3,3} as a 4-dimensional regular dodecahedron, and the 600-cell {3,3,5} as a 4-dimensional regular icosahedron. The kind and number of a cell, face, edge, and vertex of each are compiled in Table 4. The number of each element which fits together around every vertex is given in parenthesis in each case (Coxeter 1973). {3,3,3}

{4,3,3}

{3,3,4}

CELL

{3,3},5(4)

{4,3},8(4)

FACE

{3},10(6)

EDGE

10(4)

VERTEX

5

{3,4,3}

{5,3,3}

{3,3,5}

{3,3},16(8)

{3,4},24(6)

{5,3},120(4)

{3,3},600(20)

{4},24(6)

{3},32(12)

{3},96(12)

{5},720(6)

{3},1200(30)

32(4)

24(6)

96(8)

1200(4)

720(12)

24

600

16

8

Table 4: Numerical data of regular polychora.

120

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47

5 ORTHOGONAL 3-PROJECTION AND ROTATIONAL SYMMETRY OF REGULAR POLYCHORA Figure 5 compiles systematized orthogonal 2-projections of regular polyhedra showing the degree of rotational symmetry. Each is embedded in a regular polygon or a √2 rectangle. The degree of rotational symmetry is shown by nF✕m when m numbers of nfold regular polygonal symmetry can be seen. According to the difference between these degrees, they are classified into 3 groups; the tetrahedral group T of {3,3}, octahedral group O of {4,3} and {3,4}, and icosahedral group I of {5,3} and {3,5}. Figure 6 shows, on the analogy of Figure 5, the rotational symmetry of regularly chosen orthogonal 3-projections of regular polychora. Each polychoron is shown by a pair of an edge pattern and a face model. Each has rotational symmetry represented by the number of occurrence of the corresponding regular polyhedral group T, O, I, and the dihedral group Dn which is the symmetry of a regular n-gonal prism. According to the difference of the degree, they are classified into 4 groups; the 5-cell group of {3,3,3}, the 16-cell group of {4,3,3} and {3,3,4}, the 24-cell group of {3,4,3}, and the 600-cell group of {5,3,3} and {3,3,5} (Miyazaki, 2005, 2020).

{3,3}

{4,3}

{3,4}

{5,3}

{3,5}

{3},3F✕4

{4},4F✕3

{6},3F✕4

{10},5F✕6

{6},3F✕10

{4},2F✕3

2F✕6

2F✕6

{4},2F✕15

{4},2F✕15

{3},3F✕4

{6},3F✕4

{4},4F✕3

{6},3F✕10

{10},5F✕6

Facecenter

Edgecenter

Vertexcenter

Figure 5: Orthogonal 2-projections with a degree of symmetry of regular polyhedra.

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K. MIYAZAKI, M. ISHII

{3,3,3}

{4,3,3}

3,3,4}

{3,4,3}

{5,3,3}

{3,3,5}

T✕16

O✕24

I✕120

T✕600

Cellcenter

T✕5

O✕8

Facecenter

D3✕10

D4✕24

D3✕32

D3✕96

D5✕720

D3✕1200

Edgecenter

D3✕10

D3✕32

D4✕24

D3✕96

D5✕1200

D3✕720

T✕5

T✕16

O✕8

O✕24

T✕600

I✕120

Vertexcenter

Figure 6: Orthogonal 3-projections with a degree of symmetry of regular polychora.

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6 ORTHOGONAL 2-PROJECTION AND ROTATIONAL SYMMETRY OF REGULAR POLYCHORA Polyhedra as projections into 3-space of polychora can again be projected onto a 2-plane as polygonal patterns which construct plan-elevation pairs of original polyhedra. Figure 7 shows the case of {5,3,3} (left half) and {3,3,5} (right half) of a vertex-, edge-, face-, and cell-centered projection from left to right. Their oblique views are arranged in Figure 6. Each of the polygons seen in plan-elevation pairs represents one of the following 3 patterns: ■ type having regular polygonal 2-fold (or 4-fold), ▲ type of 3- (or 6-) fold, and ● type of 5- (or 10-) fold rotational symmetry. Especially, triplets of a plan, elevation, and special side-view of cell-centered projection of {5,3,3} and vertex-center projection of {3,3,5} are composed of ●(plan), ■(elevation), and ▲(special side-view) types as shown in Figure 8. The 4-dimensional coordinate axes shown in Figure 3 are depicted inside. Furthermore, congruent ■ types appear in orthogonal 2-projections onto XY, YZ, ZU, and UX plane of each regular polytopes as shown in Figure 9 (Miyazaki, 2005, 2020). Figure 10 shows the edge-patterns of orthogonal 2-projections (upper) and face-models of 3-projections (lower) of regular polychora in general position. Each of the projections is embedded in a regular polygon having as many edges as possible. They cannot be seen in Figure 6, and none of them is cell-, face-, edge-, nor vertex-centered (Banchoff,1990).

Figure 7: Plan-elevation pairs of polyhedra as orthogonal 3-projection of {5,3,3} (left half) and {3,3,5} (right half).

Figure 8: Triplets of plan (●), elevation (■), and special side-view (▲) of {5,3,3} (left) and {3,3,5} (right).

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Figure 9: 4-fold symmetric patterns derived from orthogonal 2-projections onto XY, YZ, ZU, and UX planes of {5,3,3} (left) and {3,3,5} (right).

{5},{3,3,3} {8},{4,3,3} {8},{3,3,4}

{12},{3,4,3} {30},{5,3,3}

{30},{3,3,5}

Figure 10: Projections of regular polychora in general position.

7 PLANAR TESSELLATIONS BY ORTHOGONAL 2-PROJECTION OF REGULAR POLYCHORA Infinite copies of a suitable polygon as orthogonal 2-projection of regular polychora can tessellate a 2-plane. For example, a regular triangle, square, and regular hexagon seen in Figure 6 can construct periodic Pythagorean regular tessellations, as shown in Figure 11. a is derived from {3,3,3}, b from {4,3,3}, c from {3,3,4}, and d from {3,4,3}. Furthermore, copies of a regular pentagon, octagon, and decagon can construct aperiodic tessellations as shown in Figure 12 (Grünbaum, 1987). A regular pentagon is also derived from orthogonal 2-projection of {3,3,3}, a regular octagon from {4,3,3}, {3,3,4}, and {3,4,3}, and a regular decagon from {5,3,3} and {3,3,5}. Figure13 shows aperiodic tessellations by these 2-projections. Each unit is shown at each upper left corner. They can also be seen in Figures 7 and 10 except {3,3,3} and {3,4,3}. 2-projections of golden heptahedra found by Quehenberger as a 3-dimensional representation of the Penrose Kites & Darts tiling can be embedded in the cases by decagons (Quehenberger, 2014).

SYMMETRY IN PROJECTION OF 4-DIMENSIONAL REGULAR POLYCHORA

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Figure 11: Regular polygons as 2-projections of regular polychora which can construct Pythagorean regular tessellations.

Figure 12: Aperiodic tessellations by regular pentagons, octagons, and decagons devised by J.Kepler, R.Ammann, and R.Penrose from left to right.

Figure 13: Aperiodic tessellations by orthogonal 2-projections of regular polychora.

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8 ONE-POINT PERSPECTIVE AND ROTATIONAL SYMMETRY OF REGULAR POLYCHORA Orthogonal projection can be assumed to be a one-point perspective whose vanishing point is at infinity. Their shapes slightly differ from each other, but their symmetry properties are the same. Figure 14 shows edge-patterns and face-models of a one-point perspective of cell-centered regular polychora each of whose orthogonal 3-projection is shown in Figure 6. The 4-dimensional analogue of the Schlegel diagram of a 3-dimensional polyhedron represents a special example of one-point perspective as shown in Figure 15. Its outside shows a regular polyhedral cell of any of regular polychora and inside the other cells showing a one-point perspective of the polychoron. Each of them except {3,3,5} has the same rotational symmetry as the polychora of Figure 14. In the case of {3,3,5}, the diagram can take 2 shapes - as can be seen at the 2 right corners. One is in a regular tetrahedron and the other in a regular icosahedron because {3,3,5} is composed of 600 tetrahedra, and also of 24 icosahedra and 120 tetrahedra. The rotational symmetry of this icosahedron is not I but T for its inside structure. If this icosahedron is added, all of the 5 kinds of 3-dimensional regular polyhedra occur. Therefore, every composition by regular polyhedra in 3-space can be reproduced in 4-space by using them, though there are 3 kinds of a regular tetrahedron and symmetry of an icosahedron differs.

{3,3,3}T

{4,3,3}O

{3,3,4}T

{3,4,3}O

{5,3,3}I

{3,3,5}T

Figure 14: Edge-patterns and face models of a one-point perspective of cell-centered regular polychora.

{3,3,3}T {4,3,3}O {3,3,4}T

{3,4,3}O

{5,3,3}I

{3,3,5}T

Figure 15: The 4-dimensional analogue of the Schlegel diagram.

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9 ORTHOGONAL 3-PROJECTION AND REFLECTIONAL SYMMETRY OF SEMIREGULAR POLYCHORA In 3-space, there are 13 kinds of Archimedean semiregular polyhedra. All of them are classified into 3 kinds of regular polyhedral groups according to rotation symmetry. Of them, 11 kinds except two of snub types and all of the regular polyhedra have reflection symmetry and can be designed by Wythoff construction using the symmetry operation. Figure 16 shows the case of 7 kinds of the icosahedral type. The left part shows a spherical 120-hedron whose edges are great circles embracing a spherical regular icosahedron and dodecahedron. The face is a scalene triangle having 3 vertices A, B, C, 3 edge-centers D, E, F, and a face-center G. When these 7 points are reflected across all of the edges, the 7 kinds of polyhedral are obtained. Each of them can be arranged at 7 vertices of a cube and has coordinates whose origin is at the center of the 120-hedron as one of the vertices of the cube. The coordinates can also be represented by a Coxeter-Dynkin diagram as one of Coxeter group of degree 120. According to similar construction, the tetrahedral and octahedral group belongs to the Coxeter group of degree 24 and 48, respectively. The 4-dimensional analogue for the group of {3,3,5} is shown in Figure 17. All 15 kinds of regular and semiregular polychora are obtained. The 4-dimensional analogue of a spherical 120-hedron is a 4-dimensional spherical 14400(=24×600)-choron made of tetrahedral cells. The tetrahedron has 4 vertices, 6 edge centers, 4 face centers, and a body center. When these 15 points are reflected across all of the faces, all 15 kinds are obtained. Each of them can be arranged at 15 vertices of a 4-cube and has coordinates whose origin is at the center of the 14400-choron as one of the vertices of the 4-cube. The coordinates can also be represented by a Coxeter-Dynkin diagram as one of the Coxeter group of order 14400. Similarly, the order of the Coxeter group of {3,3,3}, {3,3,4}, and {3,4,3} is 120, 384, and 1152, respectively (Coxeter 1973, 1974). H. Lalvani originally devised a similar construction for various geometrical figures (Lalvani, 1982).

10 SPHERICAL STRUCTURES ON THE MODEL OF SEMIREGULAR POLYCHORA Semiregular polychora are classified into 2 types. One is a single type which is derived from regular polychora by truncation applied once, and the other of a double type derived by truncation applied twice.

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Figure 16: Wythoff construction of icosahedral polyhedra. Coxeter-Dynkin diagrams and symbols of regular and semiregular polyhedral cells are attached.

Figure 17: Wythoff construction of polychora derived from {3,3,5}.

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In the case of the group of {3,3,5}, the single type is composed of 2 kinds of regular or semiregular polyhedral cells, and the double type of 3 or 4 kinds including regular prisms. Of their projections, double-types show complicated spherical polyhedra each of whose inside is packed by many distorted regular or semiregular polyhedral cells. They are available for design of large, stable, and variable space structures, especially spherical domes as shown in Figure 18. Upper of each shows an oblique drawing, and lower planelevation pairs derived from truncation of cell-, face-, edge-, and vertex-center projection of {3,3,5} from left to right. Each plan shows ●,▲, or ■ type pattern, and elevation a half part of ■ type to stand against gravity. These domes can cover the city plan with an aperiodic pattern as shown in Figure 13 (Miyazaki, 2005, 2020, Robbin, 1996, 2006).

Figure 18: Architectural drawings of semiregular polychora belonging to the double-typed {3,3,5} group.

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11 CONCLUSIONS 4- or higher-dimensional shapes are usually analyzed and illustrated in an abstract treatment by using complex and high-level numerical theories so that it is difficult to use them practically. On the contrary, when they are represented by using simple and primitive graphical tools, their appearance becomes concrete and applicable in art and science. As suitable tools, this paper introduced mainly basic orthogonal and one-point perspective projection, rotational and reflectional symmetry, and regular polychora. As a result, for example, some of the exotic aperiodic planar patterns shown in Figure 13 and useful spherical patterns shown in Figure 18 were derived. All are consequences of symmetry properties.

BIBLIOGRAPHY Banchoff, T.F. (1990) Beyond the Third Dimension, Freeman, New York. Coxeter, H.S.M. (1973) Regular Polytopes (third edition), Dover, New York. Coxeter, H.S.M. (1974) Regular Complex Polytopes, Cambridge U.P., Cambridge. Cromwell, P.R. (1997) Polyhedra, Cambridge U.P., Cambridge. Darvas, G. (2007) Symmetry, Birkhäuser, Basel. Grünbaum, B, Shephard, G.C. (1987) Tilings and Patterns, Freeman, New York. Lalvani, H. (1982) Structures on Hyper-Structures, Haresh Lalvani, New York. Miyazaki, K. (1986) An Adventure in Multidimensional Space, Wiley, New York. Miyazaki, K., Ishii, M., Yamaguchi, S. (2005) Science of Higher-Dimensional Shapes and Symmetry, Kyoto U.P., Kyoto (in Japanese). Miyazaki, K. (2020) Encyclopedia of Four-dimensional Graphics, Maruzen, Tokyo (in Japanese). Quehenberger, R. (2014) A newly found golden heptahedron named epitahedron, Symmetry: Culture and Science, 25, 177-192. Robbin, T. (1996) Engineering a New Architecture, Yale U.P., New Haven. Robbin, T. (2006) Shadows of Reality, Yale U.P., New Haven.

Morphic Polytopes and Symmetries Guy Inchbald

Independent. Park View, Queenhill, Upton on Severn, Worcestershire WR8 0RE, UK. E-mail: [email protected]; http://www.steelpillow.com/polyhedra.

Abstract: The study of symmetry began with polygons and polyhedra. Modern theories distinguish abstract and geometrical symmetries, but the relationships between these remain erratic or obscure. The morphic theory introduces an intermediate level of topological expression which helps to clarify these relationships. From this, it leads to new regular polytopes. It also offers a new approach to compound polyhedra. Keywords: symmetry, complex, simple, polygon, polyhedron, polyhedra, polytope, abstract, morphic, geometric, concrete, topology, manifold, graph, regular, compound.

1 POLYHEDRA AND TOPOLOGY 1.1 Geometric polytopes Many of our earliest mathematical ideas of symmetry come from the regular polygons, which exhibit complex rotational and reflexive symmetries. Their use as tiles for wall and floor added translational symmetries. The regular convex or Platonic polyhedra were among the first to be discovered in ancient times and were studied principally for their symmetries. Archimedes gave his name to the slightly less symmetrical convex semiregular solids, of interest because they still share the same symmetry transformations as the regular ones. Already we see symmetries arising as a unifying mathematical idea, if only across the polytopes which exhibit them. Studies of such polytopes (generalizations of polygons and polyhedra to any number of dimensions) still often focus on their symmetries. Indeed there can be a tendency to slip into the study of symmetries without paying much attention

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Darvas (ed.), Complex Symmetries, https://doi.org/10.1007/978-3-030-88059-0_6

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to the fact that one has left the polytopes themselves behind. For example, Coxeter was apt on occasion to take the discussion into areas more appropriate to projective configurations, nevertheless continuing to refer to them as regular polytopes. (Coxeter, 1973 and 1974). Over a thousand years passed before Renaissance artists discovered symmetrical star polyhedra, but it was Kepler who first recognised that he had on his hands regular stars which self-intersect. When Euler took account of edges for the first time, it enabled him to study polyhedra as graphs or surface decompositions and thus create the discipline we now call topology. Schläfli unified polygons, polyhedra and higher-dimensional analogues in his polytopes. For over two thousand years a polyhedron had remained a solid. But from the moment when Euler noticed the equivalence with graphs, the focus moved more and more to the surface until the polyhedron became defined as its surface and its body discarded as of no significance. Topology initially treated a polyhedron as a smooth, unbounded surface or manifold, decomposed into individual polygons as a 3-connected graph. It expanded to admit higher dimensions and other kinds of an object including bounded as well as unbounded manifolds. Although the two disciplines of polytopes and topology began to diverge, they remained tightly connected across their common ground. And yet it seems not to have been until the 1950s that Stewart (1964) actually defined a polyhedron as a connected, unbounded manifold which has been decomposed into polygons. Mathematical symmetries eventually appeared elsewhere, for example with Felix Klein's Erlangen Program building all of geometry from the abstract symmetries of group theory. Through the twentieth century, a more abstract combinatorial approach developed, in which a polytope was defined according to the combinatorics of its vertex adjacencies. More generally incidence complexes, and especially CW complexes, arose as less structured assemblages of geometric elements. For these among other reasons the theories of symmetry and polytopes began to drift apart just as topology and polytopes were doing, though writers such as Coxeter instinctively continued to gloss over the consequences. Set theory enveloped both the combinatorial and incidence approaches. But then regular abstract polytopes came under attention and common ground was sought once again. (McMullen, 1997, 2002)(Johnson, 2018)

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1.2 Abstract polytopes and realization An abstract polytope is a lattice forming a partially-ordered set (poset) of ranked elements or faces, with a pairwise incidence relation between elements of adjacent rank. (McMullen, 2002) All j-dimensional faces or j-faces of an n-polytope (for −1 ≤ j ≤ n) are placed in the same rank j. Its incidence relations may be depicted in a Hasse diagram, which shows the incidences between the ranked faces as lines and the faces as nodes. Figure 1(a) is a Hasse diagram of the cube.

Figure 1: Directed Hasse diagram (a) and two realizations (b), (c) of the abstract cube

The abstract polytope has no geometric form until mapped or injected into some containing space, a process known as realization, as in Figure 1(b). But the form of the resulting geometrical figure might not look much like the abstract structure. The hourglass-like polyhedron in Figure 1(c) has four cross-quadrilateral faces and four edges geometrically incident at a false vertex but not structurally incident, which is therefore said to render it "unfaithful." The accurate cube is then a faithful realization. The awkwardness of introducing unfaithfulness is compounded by valid abstract polytopes which have no unique associated manifold because, unlike graphs or CW complexes, their j-faces are not necessarily simple topological j-spheres and so the usual topological analysis of the surface can not be applied. What does faithfulness mean for these polytopes? More generally, which mappings of which polytopes preserve faithfulness and which do not? The questions mean nothing while faithfulness remains undefined. Indeed the whole idea of realization has turned out to be just a woolly rag-bag of all the problems, old and new, that had grown up over the years and abstract theory had hoped to resolve. Clearly, there is a theoretical gap to be filled.

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1.3 Morphic polytopes In order to approach the problem of realization, I have developed a topological approach which I call morphic theory. In this, I must acknowledge much support and help from the late Norman Johnson and Branko Grünbaum. This is the first appearance of morphic theory in print but it has had to be heavily condensed. Its terminology is defined where it first appears. More appears on my website. (Inchbald 2020) The topological approach is in itself not new, being Euler's original "stereometry" much expanded by Henri Poincaré as "analysis situs" and set on an algebraic footing by Emmy Noether and others. Morphic theory picks up on Stewart's topological definition of polyhedra but goes one step further, by considering not only the faces as an unbounded topological surface but also the interior or body as a smooth topological space of one dimension higher and bounded by the subdivided surface. Thus to completely describe a polyhedron or any other polytope, it is necessary to also define the topology of its interior. The various ranks of an abstract polytope have no corresponding geometrical meaning until one chooses them. David Hilbert famously remarked in similar circumstances that rather than points, lines and planes one might equally well talk abstractly of tables, chairs and beer mugs. The choice of orientation in which to order the ranking is therefore a necessary step in the realization process. Morphic theory lends itself to an understanding of realization as a two-step process. Grünbaum (2006) identifies each j-rank with the j-faces of some graph or cell complex. The first morphic step, of interpretation, relaxes this to accommodate the non-ball j-faces of certain abstract polytopes. Instead, the inner topology of each j-face must be individually specified. I call such a complex a morphic complex. We name its cells as vertices, edges, faces, ..., body and the abstract incidence relation as one of graph connectivity. The assembled cells form a smooth bounded manifold, a morphic polytope. There remains a bijection between the abstract polytope and its morphic realization. Skeleta with hollow faces and other such oddities are not morphic polytopes. The second step in realization is then to immerse or inject the bounded manifold, complete with a morphic complex, in some containing space. The result is a geometric polytope. Because a morphic polytope is already in some fundamental sense geometric (in the same way as projective geometry), a particular example of such an immersion is called here a concrete polytope. If a direct bijection with the abstract form is obtained then the injection is an embedding. Some abstract theorists regard this as a sufficient criterion for faithfulness, but there are still problems. It may not be possible to associate a manifold

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which can be embedded, and even if it is possible one may still wish to twist it up in some way, like for example with a regular star polygon, as an immersion. 1.4 Non-simple cells Abstract polytopes incorporating non-ball j-faces such as projective planes or toroids have no counterparts in traditional definitions of a polytope. Since they are no longer CW complexes, basic tools such as Euler's formula and its higher-dimensional derivatives no longer yield topological information about them. Yet they arise so naturally from the elegant defining properties of abstract polytopes, and are so elusive in their ability to be readily identified by any abstract properties, that they have become accepted by the abstract community as valid polytopes. To obtain a morphic polytope from one, it is necessary to identify an interior or body for each cell. This cell body becomes the corresponding region of the associated manifold and must also be non-simple; inside a bounding projective plane, it must have a nonorientable twist, inside a toroid it must have a hole through it. But why stop there? The morphic approach does not bar the possibility of having a nonsimple body within a simple boundary too. This is the crucial insight which morphic theory brings to the realization of abstract polytopes. I share its origins with Grünbaum. Consider a Möbius strip as a topological surface and subdivide its boundary into five equal 1-cells or edges. Abstractly this is just a pentagon. But flattening it down into the plane and forcing an extra half-twist or two brings a surprise; it becomes a star pentagon with a hollow centre, as in Figure 2.

Figure 2: Morphing a Möbius strip into a star pentagon

Topologically this star is distinct from the usual star pentagon, which is a simple disc double-wound to give a central region of density 2. The morphic theory, therefore, treats it as a distinct new polygon. I will call such a multiple-wound orientable polytope a

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Cayley polytope, after his analysis of the regular star polyhedra in this fashion. (Cayley 1859a, b) Yet both morphic polygons are realizations of the same abstract polygon. The Möbius pentagon can take other concrete forms, as for example in Figure 3.

Figure 3: Isomorphic Möbius pentagons with differeing concrete symmetries

Joining a Cayley and a Möbius star pentagon along their edges yields a dihedron, which is a valid abstract and morphic polyhedron. It is also a projective plane. For Euclidean geometers unhappy with dihedra, a ring of five quadrilaterals may be inserted to create an abstract prism (Figure 4). Although each of the two ends is a single polygon, their topologies and realizations differ; one is open in the centre, the other closed. The doublewound central region of the Cayley star is surrounded on both sides by external space, forming a standalone membrane which does not enclose any interior; the density of the space on either side is 0. The polyhedron as a whole is non-orientable, therefore a consistent density of 2 cannot be ascribed to the membrane; the best one can achieve with such surfaces is a parity value of 0 or 1 and the double-wound region has parity 0. Should the membrane be there or not? The morphic theory provides a clear and unambiguous understanding; the usual models of density counts and Euler value do not apply to polytopes with non-orientable cells, the polygon is a double-wound disc and therefore the membrane must be Figure 4: A polyhedral projective plane present, despite its zero parity. 1.5 Overlays and twists When a morphic polygon is flattened down into the plane, multiple local regions of the manifold may overlay each other in multiple layers. Any twists in the surface will collapse to singularities where the surface crosses over or winds around itself. The central region of the Cayley star pentagon is an overlay, while the five crossing points (false vertices) of the Möbius star pentagon are twist singularities.

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The reverse process can also be applied. Given that the two star pentagons have different local topologies at the crossings, by varying these local topologies we can change the topology of the interior manifold to create different morphic polygons. But such variations do not necessarily lead to new manifolds. The dodecagon in Figure 5 is morphed in two different ways. The twist in each resulting morph occurs in a different place and the overlays also differ, however they still share the same topology.

Figure 5: Two ways to morph a dodecagon, yielding congruent boundaries.

Thus for a given abstract polytope with a given concrete boundary, it may be given different morphic bodies, while any given morphic body may take different isomorphs.

2 ABSTRACT, MORPHIC AND CONCRETE SYMMETRIES 2.1 Isomorphisms as symmetries Noether's theorem states that every differentiable symmetry of physical action is equivalent to a conservation law. This direct equivalence of conservation and symmetry is a mathematical one. It applies to more than just physics. Any isomorphism is a conservation of structural form and therefore also a symmetry. A bijective mapping between abstract and geometric forms establishes an isomorphism between them. Not all concrete realizations are bijective, so bijectivity became a fundamental requirement for faithfulness. All morphic realizations are bijective. There is thus an isomorphism, a symmetry, encompassing the abstract, morphic and faithful concrete

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examples. This symmetry between geometric isomorphs is a purely abstract one, distinct from any internal symmetries of a given polytope. The duality of polyhedra - the exchange of vertices with faces - has been remarked on since ancient times. But only with the rise of abstract symmetries could duality be understood in itself to be a fundamental symmetry. A duality of points and planes arises as a theorem of projective geometry in three dimensions, the same geometry which Klein built his program on. Like translation, rotation and reflection, duality is also a projective transformation. Such transformations all conserve certain properties of the object transformed and are hence symmetries. We may define a vertex as a (solid) angle of lines and a face as a region of points. For convex polyhedra, polyhedral and projective dualities then become indistinguishable. The parallel breaks down for non-convex figures, where projectively polarising a finite polyhedron may no longer yield a finite dual. (Wenninger 1983) Projective geometries in n dimensions have similar dualities which are reflected in n-polytopes. In theoretical physics, dualities of various kinds between theoretical models are commonly understood as symmetries. The idea or "schema" of an underlying common mathematical core to an isomorphic pair of dual theories or models has been proposed. (De Haro, 2019) This schema approach suggests the idea of an abstract polytope as a common core underlying its dual realizations. In abstract theory, the duality of polytopes manifests in a remarkably beautiful way. To obtain the dual of a polytope we need only turn its Hasse diagram upside down so that the direction of the ranking order is reversed (Figure 6). The internal symmetries of the polytope are unaffected.

Figure 6: The directed Hasse diagrams of the triangular prism (a) and its dual the trigonal dipyramid (b) are isomorphic but ranked in opposite directions.

At its most abstract, the set-theoretic model does not need to direct the order of ranking. A geometric polytope and its dual have identical abstract structures and may thus be

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understood as realizations of one and the same abstract polytope. It is only at the first, interpretative step of realization that a direction is chosen. 2.2 Internal symmetries and realization When the need for faithfulness in abstract realizations was first recognised, opinions differed over the need to conserve internal symmetries. A geometric parallelepiped is an embedding of an abstract cube having higher symmetry. One approach regards the parallelepiped as faithful. After all, it is a bijection, another as unfaithful because it does not preserve the internal symmetries. One may define faithfulness as suits the task at hand. In what follows, preservation of all symmetries, internal and external, is assumed. If all cells are balls, the morphic realization preserves abstract symmetries. However, with arbitrary non-ball topologies allowed, it does not necessarily do so. The pentagonal star prism in Figure 4 has an abstract symmetry between the two end faces but the projective morphic realization with one Möbius face breaks that symmetry. Making both faces hollow, like a Klein bottle, would restore the lost symmetry. Not all concretizations merely preserve or break symmetry, some create it. The symmetry of the gyrobifastidium in Figure 7(a) is a subgroup of tetrahedral. Consider a morph in which the two corners of each wedge are brought together and superimposed at a central point, such that the connected edge shrinks to zero length, as Figure 7(c). This concretization has the full symmetry of the regular octahedron. But this does not make it super-faithful; it fails to maintain the fundamental isomorphism and is therefore unfaithful.

Figure 7: Morphing a gyrobifastidium

Thus the abstract, morphic and concrete symmetries of a given geometric polytope can vary significantly from one another. However, they are almost always interdependent to some extent and so cannot be treated wholly separately.

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3 COMPLEX SYMMETRIES IN MORPHIC POLYTOPES AND COMPOUNDS 3.1 Automorphisms and kaleidoscopes Abstractly, a symmetry of a polytope is an automorphism of its incidence structure. A symmetry group is an automorphism group. Geometrically all such groups may be generated from kaleidoscopes constructed on some associated manifold. Kaleidoscopes inherently generate complex symmetries. The mirrors of the kaleidoscope individually generate reflective symmetries, while the (rational) angles between them generate rotational symmetries. Parallel mirrors spaced rationally instead generate translational symmetries (and replicate centres of rotation). In general, a mirror on an n-manifold may be of any dimension from 0 to (n−1) and reflection is often called inversion. Simple symmetries (such as chiral) may be seen as closed subsets of the full kaleidoscope. All these symmetries may be classified by their orbifolds. (Conway, 2008) A simple reflection or inversion is a twofold symmetry. A simple rotational symmetry may be of greater order. The smaller the angle between mirrors, the larger the modulus of the rotation group, with a zero angle corresponding to simple translation. If the associated manifold is finite then both rotation and translation moduli will be finite. This is seen for example in the {4, 4} tilings of the torus. Infinite associated manifolds include spaces and sponges, yielding respectively tilings or tessellations and infinite polyhedra or polyhedral sponges. (Wachman et al, 2005, Burt, 2006) Other automorphisms, such as glide reflection, are complex symmetries. 3.2 Regular morphic polytopes A polytope is regular if its automorphism group acts transitively on its flags. For a morphic polytope, an automorphism also requires that the corresponding cell bodies be homeomorphic. For a concrete polytope, an automorphism further includes the arrangements of overlays and twists. The Cayley and Möbius star pentagons described are both regular. The hexagon can be treated similarly, however, its star realizations are less faithful to its abstract structure. Grünbaum (1993) described a skeletal double-wound hexagon {6/2} whose edges and vertices coincide to give the appearance of a triangle. If we realize the body as a disc, it is double-wound with density 2. Doing the same with a Möbius strip results in a body

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trapped between coincident edges and of zero area, with the central void expanding to occupy the entire bounded area (Figure 8).

Figure 8: Flatteining a Möbius hexagon

Clearly, every n-gon can be treated in this way with a Möbius body, yielding an entire family of new regular polygons; some will have finite bodies, some will not. Figure 9 shows four regular hexagons; two old examples and two new morphic polygons. Figure 9: Some regular hexagons ABCDEF; Convex, Cayley, Möbius and skew Möbius

The principle extends to polygons with more general interiors including toroidal handles. Such a polygon will be isomorphic to some higher toroidal surface with a hole cut out to create a boundary. The first new 11-gon in Figure 10 is a Möbius strip, the second a more complicated manifold. In fact, the second is not yet fully defined, as different arrangements of twists across each join allow various bounded toroids to adopt this pattern of layering. Figure 10: Two {11/4}regular morhic stars.

Investigations to date suggest that all such star toroids must be non-orientable, as the necessary conditions for an orientable figure appear also to require the boundary to be a compound figure. But this remains a conjecture. Extending the principle to three and four dimensions yields new regular star polyhedra and polychora, all sharing the same bounding surfaces as the four Kepler-Poinsot and twelve Hess solids. There are no regular stars in higher dimensions. (Coxeter, 1973)

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Although usually not regular in themselves, many stellations of the regular polytopes can now be understood as twisted manifolds, typically toroidal. By duality, the facettings of the regular polytopes are alternative realizations of the same polytopes.

Figure 11: Stellated icosahedon Ce2; (a) model in the Cambridge University Mathematics Library, possibly by. H.T. Flather, (b) face. Excluded from The Fifty-Nine Icosahedra,.it is however a valid morphic polytope.

Regular polytopes and tilings are often identified by their Schläfli symbols. However, the identification is not necessarily unique. For example {4, 4} represents the plane tiling of squares. But it may also be constructed on other manifolds, such as a toroid of genus 1, with many variations in the number of tiles. The translational symmetries, and hence groups of these tilings may differ. Thus, a Schläfli symbol does not identify a unique polytope or even a unique symmetry group; it is little more than a rotational construction recipe which, if followed, will yield a regular figure. Morphic theory expands the range of regular polytopes which may be constructed according to any given Schläfli symbol. 3.3 Compounds as polyfolds The morphic theory allows a novel treatment of polyhedral compounds. We have seen that {6/2} sometimes denotes a double-wound triangle, as shown in Figure 12(a). The abstract symmetry is sixfold but the geometry of this concrete realization is threefold. Therefore the symbol is often taken instead to denote a regular compound of two triangles, the hexagram in (b). But, while this concretizes the desired symmetry, it divides up the structure into two triangles thus breaking its abstract symmetry. If we instead treat the compound figure topologically and hence abstractly as a single 2-manifold with a boundary of two disconnected pieces, we may then regard the regular star polygon as a single abstract figure (c). In general, the number of boundaries is denoted by the highest common factor between the numerator and denominator of the Schläfli symbol. The abstract structure no longer conforms to any definition of a polytope and represents a slightly more general class of incidence complex and associated manifold

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having their own symmetries. Here, I will refer to this class as polytopal manifolds, or polyfolds (they remain a sub-class of CW complexes).

Figure 12: Hasse diagrams of {6/2} as; (a) a double-wound Cayley or Möbius hexagon, (b) two discrete triangles and (c) a manifold with two triangular boundaries

The abstract and morphic symmetries of the double-wound triangle (b) are those of the regular convex hexagon, while its concrete symmetry is only threefold. The doublebounded star (c) shows a different abstract structure which yields different permutations of flags – and hence different symmetries – from the abstract hexagon, yet its concrete symmetries are those of the regular convex hexagon. Which of them represents {6/2} must still depend on whether we are considering the abstract or concrete symmetry, and if we choose the star then we must still accept that {6/2} is not a polytope. One might distinguish between a "polygonal compound" such as two overlapping but discrete triangles on the one hand, as in (b), and a "compound polygon" as a manifold with two triangular boundaries on the other, as in (c). The same principle of a polyfold having a single body may be readily extended to compounds of the regular polyhedra and polytopes in general, with a compound of c polytopes being understood as a manifold with c boundaries. The general class of polyfolds may then be said to comprise both solo and compound polytopes, and therefore any Schläfli symbol will represent one or more regular polyfolds. Compounds can be used to break down complex symmetries into their simpler components. For example, the regular compound of five tetrahedra is chiral, having only

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rotational symmetry. Compounding it with its handed twin yields the regular compound of ten tetrahedra, which has complex (rotational and reflexive) symmetries. Since duality is a symmetry, a compound of two dual polytopes exhibits this symmetry. Unless it is in some way distorted, it will also exhibit the symmetries of the original polytopes. Thus the symmetry of a dual compound is of a higher complex order than that of the original polytope. The regular tetrahedron is self-dual. The regular compound of two tetrahedra or stella octangula exhibits not only the complex geometric symmetry of the original regular tetrahedron but also, due to its geometric arrangement, that of the regular octahedron and also that of the duality of the two parts.

REFERENCES Burt, M. (2006) Polyhedral Sponge Structures: New Imagery and Inspiration for Innovative Space Structures, IASS conference, Beijing. Cayley, A, (1859a) On Poinsot's Four New Regular Solids, Philosophical Magazine, Vol. 17, 123-128. Cayley, A, (1859b) Second Note On Poinsot's Four New Regular Polyhedra, Philosophical Magazine, Vol. 17, 209-210. Conway, J., Burgiel, H. and Goodman-Strauss, C. (2008) The Symmetries of Things, Wellesley: A K Peters. Coxeter, H.S.M. (1973) Regular Polytopes, 3rd ed, New York: Dover. Coxeter, H.S.M. (1974) Regular Complex Polytopes, Cambridge: Cambridge University Press. Coxeter, H.S.M., du Val, P., Flather, H.T. and Petrie, J.F. (1938) The Fifty-Nine Icosahedra, Toronto: University of Toronto. De Haro, S. and Butterfield, J. (2019) On Symmetry and Duality, Synthese; (Special Issue, Online), https://doi.org/10.1007/s11229-019-02258-x Grünbaum, B. (1993) Polyhedra with Hollow Faces, Proceedings of NATO-ASI Conference on Polytopes: Abstract, Convex and Computational, Toronto, 1993, Dortrecht, Kluwer Academic, 1994, 43-70. Grünbaum, B. (2006) Graphs of Polyhedra, Polyhedra as Graphs, Discrete Mathematics, No. 307, 2007, 445-463. Inchbald, G. (2002) In Search of the Lost Icosahedra, The Mathematical Gazette, Vol. 86, No. 506, July 2002, 208-215. Inchbald, G. (2020) Morphic Polytopes, http://www.steelpillow.com/polyhedra/morphic/morphic.html Johnson, N.W.; (2018) Geometries and Transformations, Cambridge: Cambridge University Press. (See especially Chs 11 to 13). McMullen, P. and Schulte, E. (1997) Regular Polytopes in Ordinary Space, Discrete and Computational Geometry, 17, 1997, 449-478. McMullen, P. and Schulte, E. (2002) Abstract Regular Polytopes, Cambridge: Cambridge University Press. Stewart, B. (1964) Adventures Among the Toroids, Okemos: 2nd ed., Stewart. (1st ed. 1952). Wachman, A., Burt, M. and Kleinmann, M. (2005) Infinite Polyhedra, 2nd edn, Haifa: Technion. (1st edn 1974). Wenninger, M. (1983) Dual Models, Cambridge: Cambridge University Press.

Inducing the Symmetries Out of the Complexity: The Kepler Triangle and Its Kin as a Model Problem Takeshi Sugimoto

Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa Ward, Yokohama 221-8686, Japan E-mail: [email protected]

Abstract: This essay is intended to exemplify how to solve a complex problem with several symmetries. The sample problem is a generalisation of the Kepler triangle. In 1597 Kepler discussed the golden-ratio related right-triangle, which is embedded in the cross-section of the Pharaoh Khufu’s pyramid. Three hundred years have passed since the first report, there are only three ‘such’ specimens known to mankind in the 21 Century. But the author broke through the solution space to reveal the three aspects: the golden ratio, the Fibonacci sequence and the Pythagorean Theorem. The path to the solution is explained from the starting point of collecting the data, by way of finding the invariants and to the goal of deriving the laws governing the solution space. Keywords: Metallic Means, Generalised Fibonacci Sequences, Pythagorean Theorem.

1 INTRODUCTION The word ‘symmetry’ derives from ‘syn (total)’ and ‘metry (measure)’. Therefore, a symmetry is a total measure representing a geometry under consideration. The choice of a measure is object-oriented and depends on the structure of a problem. In this essay I shall show a typical process of geometry-related problem-solving: that is the path of analysis from the starting point of collecting the specimens (complex facts), by way of finding the invariants in the solution space and to the goal of deriving the laws governing

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the solution space. The sample problem is the Kepler triangle and its kin (Sugimoto, 2020a), i.e., a generalisation of the golden-ratio right-triangles. On 30 October 1597 Johannes Kepler (1571-1630) wrote to his mentor Prof Michael Mästlin (1550-1631) about the golden-ratio related right-triangle (Frisch, 1858, pp.34-38, as shown in Fig. 1.(a) below). Therefore, this triangle is called as ‘the Kepler triangle,’ but Kepler is not an inventor of this triangle. The Kepler triangle is hidden in the cross-section of the Pharaoh Khufu’s pyramid. That is, modern mathematicians have fabricated the scientific urban-legend: ‘the triangle in the side of the pyramid equals the square of the height.’ After some algebra, we obtain the relation related to the golden ratio , that is (half the base) : (the vertical height) : (the height of the triangle in the side) = 1 : 1/2 : , i.e., the Kepler triangle or the golden righttriangle. By the measurement, the base is 230 [m], whilst the vertical height was 146.5 [m]. By use of these values with four significant digits, (the vertical height)/(half the base) becomes 1.274, whilst 1/2 is 1.272. Indeed, the Kepler triangle is embedded in the Pharaoh Khufu’s pyramid.

2 GENERALISING THE KEPLER TRIANGLE 2.1 Collecting the Data: State Description When I determined to explore the solution space about the Kepler triangle, there were only three specimens known to date. These are shown in Fig.1. The Kepler triangle was classical and known to the world long before the abovementioned Kepler’s letter in 1597. The silver right-triangle was known by the 20th century, and Olsen told he learned this triangle from geometer and artist Mark Reynolds (Olsen, 2002). In 2002 Olsen introduced us to his right-triangle with 31/2 as its hypotenuse (Olsen, 2002). The three specimens are not enough. It was quite difficult to guess rules to draw the right triangle of this sort. What is ‘this’ sort? Is the short leg confined to unity? The long leg is presumably n/2 for a natural number n. The hypotenuse seems like m1/2 for a natural number m. There was, however, no clue to find the link between n and m.

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(c) (b) (a)

ξ2Φ

Φ  

Φ

Φ



1

1

1

ξ3Φ

Φ

Figure 1: (a) the Kepler triangle; (b) the silver right-triangle; (c) Olsen’s right-triangle.

About this problem, state description was not specific at first with the limitation of only three specimens. 2.2 Finding the Invariants: Aspect Description On 3rd May 2019, I played with algebra using  and found  = 5 + 3. Oh, it is obvious that 5( + ) = 52. Now, we have  + 2 = 52. Reading this relation as the Pythagorean Theorem, we obtain a new member to the Kepler triangle and its kin as shown in Fig. 2.

Φ 

ξ5Φ



ξ2 Figure 2: My discovery.

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At this moment I was liberated from the curse of ‘the short leg is unity.’ Now, we become aware that related natural numbers should be the Fibonacci numbers because of the relation to . At last, we have got the three aspects of the solution space: the golden ratio, the Fibonacci sequence and the Pythagorean Theorem. 2.3 Inducing the Laws: Process Description Let Fn be the nth Fibonacci number. The Fibonacci sequence is given by the recursion formula below. Fn = Fn-1 + Fn-2, (1) subject to the initial conditions F1 = F2 = 1. That is, {1, 1, 2, 3, 5, …}. Comparing four triangles in Figs. 1 and 2, we have got a clue from three triangles except the Kepler triangle: sides of triangles are related to every two Fibonacci numbers; unity is not mere unity, but it is a square-root of unity. To explain the cases of n = 1 and 2, we need to extend the Fibonacci sequence toward a negative generation. Using (1), we deduce F0 = 0 because of F2 = F1 = 1; then we deduce F-1 = 1, because F1 = 1 and F0 = 0. In case n = 1, we regain the Kepler triangle: (the short leg, the long leg, the hypotenuse) = (F-11/2, 1/2, F11/2) = (1, 1/2, ). In case n = 2, the triangle is degenerated to the segment: (the short leg, the long leg, the hypotenuse) = (F01/2, , F21/2) = (0, , ). Finally, we could induce the law from the three aspects. The generalised golden righttriangle has such a geometry that (the short leg, the long leg, the hypotenuse) = (Fn-21/2, n/2, Fn1/2). Its generic image is shown in Fig. 3 below. The Pythagorean Theorem is applicable to the newly defined triangle in the following form (the proof given in Appendix I). Fn = n + Fn-2

(2)

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Φ  n

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ඥ𝐹𝑛 Φ

ඥ𝐹𝑛−2 Figure 3: The generalised golden-ratio righttriangle (a generic image).

I point out that the Fibonacci sequence and the Pythagorean Theorem are essentially generators of growing patterns, e.g., the Fibonacci spiral and the fractal Pythagoras tree (as shown in Fig. 4, details of drawing explained in Appendix II). Both patterns look very organic because those algorithms act as laws in the growing process. Mr Fibonacci, Leonardo of Pisa, originally introduced his sequence to investigate rabbit-breeding, whilst a2 = b2 + c2 implies a conservation law of branching. (a)

(b)

Figure 4: Pattern formation: (a) the Fibonacci spiral; (b) the Pythagoras tree.

Thus, the Kepler triangle and its generalisation embed the golden ratio, the Fibonacci sequence and the Pythagorean Theorem. The formalism, shown in Fig. 3, covers three known specimens, shown in Fig. 1 (Sugimoto, 2020a).



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3 EXPLORING THE NEW LINE OF A STUDY Very recently I discovered the super set of the Kepler triangle and its generalisation (Sugimoto, 2020b), which I call ‘the metallic right-triangles.’ About the metallic means, it is advised to consult Vera W. de Spinadel’s works, for example, Spinadel (1999). Let (m) be the metallic mean of the metal number m. The metallic mean is defined as follows. (m) = m + 1/ (m). In case m = 1, we obtain the golden ratio ; in case m = 2, we obtain the silver ratio (2) = 1 + 21/2; in case m = 3, we obtain the bronze ratio (3) = (3 + 131/2)/2. There are generalised Fibonacci sequences adjoined to the metallic mean. Let Gn(m) be the nth generalised Fibonacci number related to the metal number m. Then, the generalised Fibonacci sequence of the metal number m is defined as follows. Gn(m) = mGn-1(m) + Gn-2(m). By use of (m) and Gn(m), we can define the metallic right-triangles. In case m = 1, we regain the Kepler triangle and its generalisation. I shall show the initial members of the metallic right-triangles for m = 1, 2 and 3 in Fig. 5. (c)

(b) (a) Φ1Τ2

ξ3Φ1Τ2 ሺ3ሻ Φሺ2ሻ

ξ2Φ1Τ2 ሺ2ሻ Φ 1

1

Φሺ3ሻ

1

Figure 5: Examples of the initial members of the metallic right-triangles: (a) the golden righttriangle; (b) the silver right-triangle; (c) the bronze right-triangle.

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Deep understanding of the solution space leads us to the higher dimensional exploration to the problem under consideration.

4 CONCLUSION To explore a complex problem with several symmetries, there is a possibly effective path towards the complete understanding of the solution space: (1) Collecting the data State description of the problem is the starting point of understanding the solution space. It is important to know how the concerned problem looks like. (2) Finding the invariants This is the stage of aspect description. The invariants can be found by studying raw data by correlation analysis, association analysis and so on. As aspects of the solution space are described very well, we are led to the cause-effect relations behind the problem. (3) Inducing the laws The stage of process description is the most difficult to attack. There is no general way to induce the laws governing the solution space. Try hard, and you will be rewarded. Following the manner above, I have succeeded to generalise the Kepler triangle and its kin twice (Sugimoto, 2020a & 2020b).

REFERENCES Frisch, C., ed. (1858) Joannis Kepleri Astronomia Opera Omnia, Frankfurt, Heyder & Zimmer, Vol. I, XIVpp. & 672pp. Olsen, S. (2002) The Indefinite Dyad and the Golden Section: Uncovering Plato’s Second Principle, Nexus Network Journal, 4, 1, http://www.nexusjournal.com/GA-v4n1-Olsen.html. https://doi.org/10.1007/s00004-001-0007-8 Spinadel, Vera W de (1999) The family of metallic means, Symmetry: Culture and Science, 10, 3-4, 317-338. https://doi.org/10.26830/symmetry_1999_3-4_317

Sugimoto, T. (2020a) The Kepler Triangle and Its Kin, Forma, 35, 1, 1-2. https://doi.org/10.5047/forma.2020.001 Sugimoto, T. (2020b) The Metallic Right-Triangles, Forma, in the press.

APPENDIX I To make a short cut we use the Binet formula for the Fibonacci numbers given by

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Fn = 5-1/2{n − (−)n}. To prepare the proof we need the following relation by use of the above formula for n =2 (F2 = 1). 2 − 2 = 51/2. Now we start from the right-hand side of (2). n + Fn-2 = n + 5-1/2{n-2 − (−)n-2} = 5-1/2{51/2n + n-2 − (−)n-2} = 5-1/2{51/2n-2 + n-4 − (−)n}2 = 5-1/2{(2 − 2)n-2 + n-4 − (−)n}2 = 5-1/2{n − n-4 + n-4 − (−)n}2 = Fn2. Thus, we arrive at the left-hand side of (2).

[Q.E.D.]

APPENDIX II [The Fibonacci spiral] Drawing process is based on the Fibonacci sequence in any kind of a length unit. Let a quarter-circle arc be inscribed in a square. Place two 1 × 1 squares side by side; place a 2 × 2 square upon them; Place a 3 × 3 square on their left sides; place a 5 × 5 square underneath them; place an 8 × 8 square on their right sides; place a 13 × 13 square upon them; place a 21 × 21 square on their left sides; this pattern is shown in Fig. 4(a). The Fibonacci spiral approximates the golden spiral found in plants and animals.

[/]

[The Pythagoras tree] This was invented by A.E. Bosman in 1942. Let a right triangle be with squares erected on every three sides. Place the largest square as the stem of the tree; grow smaller and similar triangles with their hypotenuses upon the smaller squares; let these smaller triangles be with squares erected on every two legs; iteration of this process leads to fractal geometry. Figure 4 (b) shows the Pythagorean tree grown by use of the right triangle with their sides, (the short leg, the long leg, the hypotenuse) = (1, 2, 51/2), up to the tenth iteration. [/]

Synchronizing the Isotropic Vector Matrix with the Stellated Vector Matrix Jim Lehman

Address: 3410 Harney St., Vancouver, WA 98660-1829, U.S.A. Email: [email protected]

Abstract: I show how the icosahedron, through face stellations, forms a complex original polyhedron I call the ICOSA-60 STELLATE (Fig. 39 (L-267)). This polyhedron forms a 4-fold & 5-fold matrix that encapsulates the prime vectors of the IVM (Fuller, 1975). The fourfold matrix accommodates the “omnisymmetry” (Edmondston, 1992) of the isotropic vector matrix (IVM) of R. Buckminster Fuller via 2-frequency tetrahedrons. The SVM matrix (Fig. 35 (L-119)) consists of icosahedrons and dodecahedrons that integrate and synchronize these two paradoxical symmetries into one unified matrix. Methods include paper folding, CAD drawing, and 3D printing. Keyword terminology1

INTRODUCTION 1.1 Phenomenon The complex shape of space is being defined here as Fuller defines spatial “omnisymmetry” (Edmondston, 1992) involving a balance of tetrahedral and octahedral symmetries to form a linear isotropic vector matrix (IVM). Within and surrounding the confines of each of these frozen in place vectors, I encapsulate them, keeping the vectors

1 The taxonomy of the figures is shown at the end of the paper. The figure order from Fig. 1 to Fig 55 is mainly based on the occurrence of images as they appear in the paper. The references shown, for example, as (L-142), (L-199) are mainly based on my CAD drawing file sequence.

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intact and inherently unchanged, to form the polyhedral stellated vector matrix (SVM) (Fig. 39 (L-267)), internalizing the linear IVM framework. Fuller describes the structure of the IVM as an energy field, positioned by spherical grouping. He then conceptually creates virtual connections between sphere centers to form virtual tetrahedrons. These tetrahedrons are then given a geometric polyhedral identity. Thus, he forms the structure of the Unified Field – The VE and Isotropic Vector Matrix. The most fundamental aspect of the VE to understand is that, being a geometry of absolute equilibrium wherein all fluctuation (and therefore differential) ceases, it is conceptually the geometry of what we call the zero-point or Unified Field – also called the “vacuum” of space. This exactly same structure is formed inside the SVM, only in a different position.

Fig. 53 (L-264) Fuller’s IVM

Fig. 54 (L-255)

Fig. 39 (L-267)

The Matrix used by the IVM and the SVM is based on an all-space filling symmetry. The IVM fills this space in a vector linear manner. The SVM fills all-space with polyhedrons. Internal to the SVM is an alternate linear IVM that is defined by the edges of tetrahedrons. Both IVM and SVM are composed of reflection and rotation symmetries, so their symmeries are complex. The SVM surrounds a linear IVM embedded inside regular pentagonal dodecahedrons. The SVM surrounds the symmetry of Fuller's IVM, which lacks 5-fold symmetry. The SVM provides 5-fold symmetry as an outer shell encapsulating a linear IVM. In summary, Fuller's IVM, when populated with polyhedrons, consists primarily of 4-fold symmetry and Lehman's SVM when populated with polyhedrons, consists primarily of 5-fold symmetry. The heretofore incommensurable 5-fold symmetry inside Fuller's IVM is seen here to occur outside the IVM. Fuller established the linear pathways formed via the centers of metaphysical spheres. Thus-the IVM has an inherent inside bias. The SVM is formed via the exterior of non-

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metaphysical sphere-like pentagonal dodecahedrons filling all-space thus positioning its IVM as an internal event to the outside surrounding event of the dodecahedron. 1.2 Simple Symmetries The simple symmetries of translation, rotation, reflection and glide reflection describe the models shown inside the IVM and SVM. These forms are fixed within an unchanging “omnisymmetry”. Any attempt to unlock the forms described would break the unchanging nature and intent of the shape of space as defined here. The symmetry characteristics of the tetrahedron2 and octahedron 3 are locked in place within the IVM. The models presented are well documented, such as the octahedron, stella octangular, 4 concave dodecahedron, 5 cube octahedron (VE) 6 , icosahedron 7 and dodecahedron (Fig. 1(L-205)). New polyhedrons are introduced such as the Lehman polyhedron and the trough icosahedron. In these exceptions, their symmetries are discussed as the forms are presented. These footnotes are added to provide a connection to existing recognized symmetry terminology. Many new terms are introduced in this paper that are outside documented usage.

2

The symmetry group of the tetrahedron S(T): A regular tetrahedron has 12 rotational (or orientationpreserving) symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation. The group of symmetries that includes reflections is isomorphic to "S"4, or the group of permutations of four objects, since there is one such symmetry for each permutation of the vertices of the tetrahedron. The set of orientation-preserving symmetries forms a group referred to as "A"4. 3 A regular octahedron has 24 rotational (or orientation-preserving) symmetries, 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, it is the poly that is dual to an octahedron. The group of orientation-preserving symmetries is S4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four pairs of opposite faces of the octahedron. 4 The stella octangula has 24 faces, 8 vertices with 3 incident edges and 6 vertices with 8 incident edges. Hence its dual solid has 24 vertices, 8 triangular faces and 6 octagonal faces. It is a truncated cube: Hence symmetry groups of the stella octangula are the same as those of the cube/octahedron. 5 Concave dodecahedron: It is also called the tetrahedral pentagonal dodecahedron because of its chiral tetrahedral symmetry. If you take a regular pentagon made of paper and fold it on any diagonal, the five edges form a concave pentagon. Twelve of these pentagons assemble into a concave dodecahedron with pyrite symmetry (George Hart). 6 Cubeoctahedron (VE): Its symmetry group B3, [4,3], (*432), order 48; T, [3,3], (O*332), order 24; Rotation group O, [4,3]+, (432), order 24. 7 A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron. The set of orientation-preserving symmetries forms a group referred to as A5 (the alternating group on 5 letters), and the full symmetry group (including reflections) is product A × Z. The latter group is also known as the Coxeter group H3, and is also represented by Coxeter notation [5,3] and Coxeter diagram.

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2 STELLATED VECTOR MATRIX (SVM) The stellated icosahedron vector matrix (SVM) is based on combining two paradoxically different 4-fold and 5-fold symmetries into one unified matrix that demands the modelling of each of their features. This matrix requires that there be no change from the prime vector pattern of the IVM when navigating through this 4-fold and 5-fold symmetry. The trough icosahedron (Fig. 2 (L-213)) is in the 3D symmetry pyritohedral group (Wikipedia). It is the unique polyhedral point group that is neither a rotation group nor a reflection group. It has two features: The first is six trough stellations Fig. 3 (L-234), Fig. 4 (L-78), Fig. 5 (L237), Fig. 6 (L-195) consisting of 6 irregular octahedrons emerging from the trough underlying twelve invisible triangular faces. This is Fuller's (1975) “jitterbug” in middle arrested rotation as an icosahedron. It unfolds an octahedron to a cuboctahedron (VE) keeping pyritohedral symmetry. Next are eight triangular faces consisting of 8-irregular stellated tetrahedrons (Fig. 7 (L218)) connecting to an outer dodecahedron. This forms the tips of a concave dodecahedron with an icosahedron core.

Fig. 8 (L-3)

Fig. 10 (L-228) Fig. 9 (L-208) Trough stellations connect and position trough icosas.

Fig. 11 (L-243)

Dodecahedrons can be positioned in 12 around one VE nesting via dodeca edges connected to the trough icosas (Fig. 11 (L-243)). A cube is located inside the dodecahedron under six dodecahedron tents and a tetrahedron is positioned inside the cube (Fig. 12 (L-102), Fig. 3 (L-234)). Additional relationships are shown in (Fig. 14 (L-83), Fig. 15 (L-191), Fig. 16 (L-238)).

Fig. 12 (L-102)

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When the dodecahedron is positioned in a 12 around one VE nesting, the internal tetrahedrons form a continuous unbroken network of 2frequency tetrahedrons that synchronize with the 2-frequency pattern of tetrahedrons in the IVM (Fig. 17 (L-111)). 

 

 

 

 

 

 

 

Fig. 17 (L-111)

The 2-frequency tetrahedron forms the stella octangula and two stella octangula placed edge to edge form a VE in the center. The VE pattern is repetitive within the IVM (Fig. 18 (L-245)). Fig. 18 (L-245)

A feature of the trough icosa matrix is the nesting of a polyhedron I call the Lehman polyhedron, referenced here as the L-poly (Fig. 19 (L-263), Fig. 20 (L-256)). The L-poly has six golden rectangle faces and eight skew triangles that accept icosas. The L-poly has octahedral symmetry. L-poly has 24 vertices, 26 faces and 48 edges. Euler: V(24)+F(26) = E(48)+2) The L-poly fills the cages Fig. 21 (L-105), Fig. 22 (L-106) formed by the trough icosa matrix. Fig. 23 (L-230) shows a VE in a cage before removing its tips to reveal an L-poly. Note that the tips of the VE must be removed for the L-poly to be contained in the cage.

  

Fig. 19 (L-263)

Fig. 21 (L105)

Fig. 23 (L230)

When the L-poly is divided into eight kernels with left/right symmetry (Fig. 24 (L-151), Fig. 25 (L-125)) by slicing to the center of the L-poly, forming kernels, and these kernels are inverted and placed in the sockets formed by the trough icosa matrix, a regular pentagonal dodecahedron is formed. Fig. 25 (L-125) shows several kernels being added forming one of the dodecahedron faces. This inversion of the L-poly creates a dodecahedron.

Fig. 24 (L-151)

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This is the only invertible (turning inside out) that I am aware of that creates a 4-fold polyhedron into a 5-fold polyhedron. The L-poly is transformed into the dodecahedron. Here eight slices are made to the L-polys exact center, then all kernels are rotated out as if making a stellation. These kernels are placed in seats formed by the trough icosahedron to complete the face of the dodecahedron. Fig. 25 (L-125)

The L-poly with  (phi) features is formed when the tips of the VE are removed. This creates on the square faces of the VE golden rectangles (Fig. 28 (L-5), Fig. 29 (L-144), Fig. 30 (L-51)) and the VE triangle faces, skew triangles. This transforms the 4-fold VE Fig. 31 (L-187) into a 4-fold phi-based L-poly as a connecting polygon.

Fig. 32 (L-50)

Fuller: “The four frequency tetrahedron inscribes an internal octahedron inscribes a skew icosahedron” (Fig. 33 (L-244)). The skew triangle on the L-poly connects to the octahedrons’ internal icosahedron, shown here, connected on the L-poly in this arrangement (Fig. 34 (L-251)).

  

Fig. 33 ( L-244)

Fig. 34 (L-251)

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3 RESULTS The following are the results recapping each symmetry. A tetrahedron is positioned inside dodecahedrons (Fig. 35 (L-119) & Fig. 36 (L-107)), forming a 2-frequency tetrahedron vector matrix which synchronizes with the IVM. Stella octangula are formed within the IVM which in turn form VE groupings within the IVM. A cube can be positioned inside and outside of each dodecahedron (Fig. 37 (L-193)) to form an all-space filling grouping of cubes. The cube outside encloses a concave dodecahedron (Fig. 38 (L-142)). The L-poly is nested in trough icosahedron matrix seats (Fig. 20 (L-256)) and is connected to trough icosahedron tips to fill all-space. Inverted L-poly kernels form dodecahedrons (Fig. 26 (L-12)). The inner core of these dodecahedrons is the stellated trough icosahedron. Concave dodecahedrons (Fig. 38 (L-142)) form face stellations, one of two icosahedron stellations. The other icosa stellation is formed by trough icosa. These two stellations (Fig. 39 (L-267)) of the icosa, which I call the ICOSA-60 STELLATE, form the complex connecting encapsulated stellated icosahedron vector matrix for this study. There are 60 exposed convex faces. The following figures show cluster views of this matrix (Fig. 40 (L-113), Fig. 41 (L-199), Fig. 42 (L-6), Fig. 43 (L-46), Fig. 44 (L-45), Fig. 45 (L-127), Fig. 46 (L-69)).

Fig. 47 (L277-1) The solid tetrahedron in the center is the SVM tetrahedron. The outer tetrahedrons are the IVM tetrahedrons. Fig. 49 (L-276) & Fig.48 (L-275) above

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The Fuller IVM and the Lehman SVM form vectors originating from different locations in and on the sphere that form tetrahedrons. How the tetrahedron is formed and where the tetrahedron centers are located are important to understanding the differences between the two matrices. 3.1 IVM Features (Fuller) 3-1-1 Tetrahedron edges The edges of tetrahedrons define the IVM. These edges are formed by connecting the centers of two spheres to form a prime vector. The prime vector forms the radials in the IVM. The center of the tetrahedron in Fuller's IVM is formed in the spaces between close-packed spheres. The significance of this becomes apparent when compared to the tetrahedron center in the SVM. 3.1.1 Sphere center The IVM radials connect to sphere centers. There is no connection to the sphere surface. 3.2 SVM Features (Lehman) 3-2-1 5-Fold Matrix The 5-fold matrix originates in the belly of the SVM tetrahedron as a stellated icosahedron (Fig. 39 (L-267)). This icosahedron forms sockets that position dodecahedrons, cubes, tetrahedrons, trough icosahedrons and the L-poly. The center of the SVM tetrahedron is the center of the sphere. 3.2.1 Offset Matrix Fig. (47 (L277-1)) The 1.0 edge of the IVM tetrahedron to the 1.0 edge of the SVM tetrahedron has offset the distance of 0.35353535353. The distance from tetrahedron vertex to vertex is 0.612372. 3.2.2 Dodecahedron inside spheres (Fig. 49 (L-276)) The dodecahedron is a stand-in for a sphere. The dodecahedron has the same 12 around one (Fig. 50 (L-257), Fig. 51 (L-261), Fig. 52 (L-38)) positioning with the same centers as the sphere. Where the sphere has no outer structure, the dodecahedron has an outer structural identity. Four vertices of the dodecahedron and its internal tetrahedron (Fig.

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48 (L-275)) make contact with the sphere surface. Radials between these points form the tetrahedron edges. 3.3 IVM/SVM Comparisons 3-3-1 Two Vector Matrices The Fuller matrix and the Lehman matrix (Fuller's IVM & Lehman's SVM) synchronize exactly together in the same time period. The vertices of tetrahedrons in the SVM intersect the IVM tetrahedron centers. This is where the two matrices lock together (Fig. 47 (L-277-1)). There are two tetrahedrons discussed. One is formed in the IVM and the other in the SVM. The edge dimensions are the same for both. The difference between them is their location and how they are formed. Both originate from close-packed spheres. Fuller's IVM connects the centers of two spheres to form a prime vector which is used to connect to four sphere centers to form the IVM tetrahedron. Lehman's SVM forms all four vectors within a single sphere by connecting to four positions on the sphere surface. These two methods form the same size tetrahedrons in different locations. This places the tetrahedron center of the IVM tetrahedron in the space between spheres and the SVM tetrahedron in the sphere center. The IVM tetrahedron center has been made impervious to change by Fuller to have fixed prime vector edges. Fuller does not allow the prime vector-based tetrahedron to be broken in such a way as to have an octahedron center. The SVM tetrahedron has at its center the stellated trough icosahedron. 3.3.1 Vector Formation The prime vector, forming the tetrahedron edge in the IVM, requires two spheres using one radius from each to form the prime vector. Note that this prime vector formation does not allow a distinction between the radius of one sphere to the next. The vector formed as a tetrahedron edge in the SVM results in having the same length as the sphere diameter. One SVM tetrahedron is formed inside a sphere and the other IVM tetrahedron is formed connecting the sphere centers of four spheres. The origin of tetrahedron edges in the IVM is from sphere centers.

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The origin of SVM tetrahedron edges inside the sphere is from 4 points on the sphere surface. 3.3.2 IVM Tetrahedron Centered in 2 Frequency SVM Tetrahedrons Each dodecahedron defines a sphere and it takes two dodecahedron centers to make a prime vector. Fuller’s prime vectors are the connections between dodecahedron centers. The endpoints of Fuller's prime vectors connect to the centers of four, 2 frequency SVM tetrahedrons. A grouping of 4 SVM tetrahedrons forms a 2-frequency edge. When the centers of those 4 tetrahedrons are connected with Fuller's prime vectors, a prime vector edge tetrahedron is formed. This tetrahedron has the same edge length as the SVM tetrahedron. Note that the tips (vertices) of the prime vectors form at the centers of the icosahedrons that are centered in dodecahedrons. The stellated icosahedron is located in the same place as the tips of Fuller's prime vectors. The centers of SVM tetrahedrons form the endpoints (vertices) of the IVM vectors. When connecting the SVM tetrahedron centers of a 2-frequency grouping of 4 tetrahedrons, a tetrahedron is formed in the center. This central tetrahedron is the prime vector tetrahedron of the IVM. The 2-frequency grouping of SVM tetrahedrons encapsulates one IVM tetrahedron. Note that the edges of each of the 4 surrounding tetrahedrons have the same edge length as the enclosed prime vector tetrahedron. 3.3.3 Two Elements of the Vector The vector has two elements: endpoints and the linear radial between the endpoints. The IVM matrix involves connecting the endpoints to the sphere center and the SVM matrix involves connecting the endpoints to the sphere surface. Each node forming the ends of Fuller's prime vectors are in the same position as the icosahedron centers formed in the SVM. These nodes are where the vertex of the IVM tetrahedron is formed.

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4 DISCUSSION Populating a matrix does not require all features to be materialized at the same time. For example, when viewing spheres that form the positioning relationships that describe the isotropic vector matrix (IVM), the spheres need not be present. It is the relationships that retain a virtual patterning that describe the matrix. Similarly, the presence of the dodecahedron containing a cube, concave dodecahedron, icosahedron or tetrahedron does not in all occasions need to be present. There is a virtual matrix pattern presence. Any time the symmetries are discussed they can populate the matrix. There is a spheric domain where connections are made with the sphere center and its surface. The SVM fully embodies the spheric domain by housing the tetrahedron with connections to the sphere center and surface. The tetrahedron formed here is being housed where 5fold symmetry resides. This is a separate isotropic vector matrix (IVM) from Fuller's IVM. While both have the exact same vector patterns, and they both evolve via the same type of sphere packing, they each are formed in different positions within the sphere packings. The SVM tetrahedrons are formed inside spheres and Fuller's IVM tetrahedrons are formed outside spheres and lie outside the spheric domain. It is the connection to the same type of sphere packing that both matrices employ that position the tetrahedron inside the dodecahedron (Fig. 35 (L-119)). IVM “bricks” include: Tetrahedron, octahedron, stella octangula, vector equilibrium (VE), cube octahedron. SVM “bricks” include tetrahedron, trough icosahedron, regular dodecahedron, concave dodecahedron, L-poly, icosahedron, cube.

5 CONCLUSION I introduce two original polyhedrons: The Lehman polyhedron (L-poly) and the Icosa-60 stellate. I show how divergent 4-fold and 5-fold symmetries cooperate in a unified stellated encapsulated matrix (SVM) Fig. 42 (L-6), each accommodating the other. By encapsulated I refer to the polyhedron based enclosures surrounding the linear matrix of R. Buckminster Fuller's Isotropic Vector Matrix (IVM) (Fig. 53 (L-264)) as a means of providing a 4-fold matrix foundation. In this matrix, 5-fold is not by nature a friendly visitor. (Fuller: Synergetics 2, 1983, 987.058): “the tetra and VE are a priori incommensurable with the icosa”. My study does not challenge Fuller's Cosmic Hierarchy

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(Fig. 54 (L-255)) or attempt to alter it in any way. The intent here is to show that simple relationships within the 4-fold symmetry of the IVM can synchronize but not be required to combine with the complex 5-fold symmetry of the stellated icosahedron vector matrix (Fig. 42 (L-6)). The symmetries of the 4-fold and 5-fold IVM and SVM do not physically merge. There is a conceptual lock-in as shown in Fig. 47 (L-277-1), but this in no way suggests that the two matrices combine. There is a synchronous, parallel, pattern where the tetrahedrons in both matrices follow the same course in an offset manner. Note that the stellated matrix of the SVM consists of tetrahedral and octahedral symmetry. Its core icosahedron follows icosahedral symmetry. There are many “bricks” that result from relationships within the IVM. The tetrahedron is formed in the SVM and the IVM. In both matrices, the tetrahedron follows the same pattern created by closest packed spheres while being offset from one another. The relationships that form “bricks” within the IVM, such as the stella octangula and the octahedron are relationships within the IVM and they are in a separate matrix from the SVM. The SVM has its “bricks” too, consisting of the concave dodecahedron and the Lpoly. Note that the relationships formed inside each of these matrices each contain a different set of “bricks”. These two matrices do not fuse together, they each contain their special set of relationships that build different polyhedrons. The concave dodecahedron, for example, does not “invade” the IVM but is associated with the IVM via the SVM matrix. The individual symmetry “bricks” in each matrix are described in detail in the reference section. I cannot emphasize enough that the tetrahedron in each matrix is formed in a different manner with respect to its origin sphere sequence. In effect, the SVM tetrahedron is formed in full, inside the sphere. The center of this tetrahedron is the center of the sphere. The IVM tetrahedron is centered on the tips of the SVM tetrahedron. As stated before, it is the edge vectors that are formed in relation to sphere centers that define the IVM tetrahedron. It is the sphere exterior, here represented by the dodecahedron, that forms the SVM vectors. It is the difference in the origin sequence for the IVM and SVM matrix vectors that determines whether the tetrahedron is formed via the sphere center or the sphere surface.

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TAXONOMY Figures not inserted in the text:

Fig. 1 (L-205) dodecahedron

Fig. 2 (L-213)

Fig. 3 (L-234)

Trough icosahedron

Fig. 4 (L-78) stellated trough

Six trough stellations

Icosa inside half dodecahedron

Fig. 6 (L-195) Stellated trough Icosahedron inside dodecahedron Fig. 5 (L-237) cross-eye stereo

Fig. 13 (L-246)

Fig. 20 (L256)

Fig. 14 (L-83)

Fig. 16 (L-238)

Fig. 15 (L-191)

Fig. 22 (L-106)

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Fig. 30 (L-51)

   

Fig. 35 (L-119) 2-frequency tet

Fig. 36 (L-107)

Fig. 37 (L-193)

Fig. 38 (L-142)

      Concave dodeca

Fig. 40 (L113)

Fig. 41 (L-199) Icosahedron stellation matrix

Fig. 42 (L-6) Stellated icosahedron Fig. 43 (L-46)

     Fig. 44 (L45)

Fig. 45 (L-127)

Fig. 46 (L-69)

Acknowledgement  to specialists who actively contributed to my research: Roger Baker: Constant support & encouragement over many years. John Braley: Contributions at Geojourney & Synergeo forums.

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John Brawley: Contributions at the Synergeo forum. Dr Scott Childs: Chemist reviewed my 4-fold & 5-fold symmetry bridge. Russel Chu: Review of my models. Mark Curtis: Correspondence on DNA. Arnie Dyer: Revising, editing & clarifying concepts. Rick Engle: Polymorph. Alan Ferguston: Collaboration on what I call the concave icosahedron. Dick Fishbeck: Frequent contributor at Geojourney. Rick Flowerday: Geometric toy designer and Synectics author. David Foster: Chairman, art Dept. U of O (deceased). Foster was the catalyst that made this research possible. R. Buckminster Fuller: I reviewed my thinking with Fuller during a personal meeting and received a letter of encouragement. George Hart: Contributed to my understanding of concave dodecahedra. Steeg Lehman: Reviewed and contributed to my studies for many years. Roy Lewallen: Analog designer. Gary Meacham: Made early photos in my studio in Eugene. Ted Murphy: Helped formulate early modelling ideas. Kenneth O’Connell: Chairman, Art Dept. U of O (retired). Rybo6: Frequent contact on the Synergeo & Geojourney forum. SketchUp: @Last Software and the SketchUp Software Team (www.sketchup.com). Kirby Urner: Webmaster of Synergetics on the Web; I Met with Kirby on several occasions to review models and my thinking. He posted on his website: The Lehman/Fuller Sculpture. Marion Walter: Geometry professor, University of Oregon.

REFERENCES Edmondston, A.C. (1992) A Fuller Explanation, Van Nostrand Reinhold, p.146. Fuller, R.B. (1975) Synergetics, Macmillian Publishing Co., Inc. Fuller, R.B. (1983) Synergetics 2, Macmillian Publishing Co., Inc. p.352; (icosa inscribed in octa), p27 (Vector Equilibrium), p.135 (IVM), p.190 (Jitterbug), p. 138 Prime vector), (“omnisymmetry”). Lehman, J. (2002) The journey: Pandora's sphere – a paradox, Symmetry: Culture and Science, 13, 3-4, 287310. https://doi.org/10.26830/symmetry_2002_3-4_287 Lehman, J. (2016) Expression – An Allegory, January 29. Lehman, J. (2016) Pandora's Sphere – A Paradox, revised and annotated, January 17. Lehman, J. (2020) The Flower of Life Enters the 4th Dimension: The Emergence of Pandora's Sphere, August. Lehman, J. © (….) Fig. 2

(L213) Trough icosahedron.

Lehman, J. © (L-poly) Fig. 39 (L-267) Icosa-60 stellate. Lehman, J. © Fig. 19 (L263) Lehman polyhedron. Lehman, J. © Fig. 38 (L142) Concave dodecahedron. Wikipedia (….) Pyritohedral Symmetry (wiki/tetrahedral symmetry).

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Patrice Jeener: Kummer's surface; it displays tetraedral symmetry. It has 16 double point contacts, 12 of which are located at the top of a truncated tetraedron and the other 4 on the top of a tetraedron.

An Analogy and Several Symmetries András Recski

Department of Computer Science and Information Theory, Budapest University of Technology and Economics, H-1521 Budapest, Hungary. E-mail: [email protected]. ORCID: 0000-0003-2371-3743

Abstract: Certain analogies between electric network theory and statics were recognized and applied some forty years ago. Using these analogies some symmetries in each area can be transformed into the other, mostly in a meaningful way. Keywords: electric network analysis, rigidity, matroids. MSC 2020: 05B35, 52C25, 70C20, 94C15 Complex symmetries usually refer to a single property of an object which is left invariant under several transformations. In this essay we consider pairs of objects (electric networks and planar bar-and-joint frameworks) connected to each other by a natural transformation and study the effect of the translation of simple symmetries in electric network theory into statics and the other way round. Our main tool will be matroid theory which, in itself, is a general common framework for many results in graph theory, in linear algebra and in geometry.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 G. Darvas (ed.), Complex Symmetries, https://doi.org/10.1007/978-3-030-88059-0_9

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1 PRELIMINARY – ELECTRIC NETWORKS Definition 1. In electric network theory a (linear time-invariant) n-port a device is described by Au+Bi=0, where A, B are k×n matrices with r(A|B)=k (the rank of the nport), and u, i are vectors of height n (whose elements are interpreted as port-voltages and port-currents). For example, the first 1-port of Figure 1 is a resistor, described by u – Ri = 0, the second and the third ones are the short circuit and the open circuit, respectively, described by u = 0 (and i is arbitrary), and by i = 0 (and u is arbitrary), respectively. The last two fictitious 1-ports are the nullator, described by u = i = 0, and the norator, described by „both u and i are arbitrary”. According to Definition 1, they are defined by the respective matrices (A|B) = 1 | 𝑅 , (A|B) = 1 | 0 ,

(A|B) = 0 | 1 , (A|B) =

1 0

0 and the 1

„empty matrix” (since we have no equations whatsoever), their ranks are 1, 1, 1, 2 and 0, respectively.

Figure 1: A resistor, a short circuit, an open circuit, a nullator and a norator

If k is the rank of the n-port, we need 2n–k further equations to determine all the port voltages and port currents. Let, say, {a1, b1, a2, b2, ..., ap, bp, ap+1, ap+2, ..., ap+q, bp+q+1, bp+q+2, ..., bp+q+r} be a maximal set of linearly independent columns (hence 2p+q+r=k). Then the corresponding values u1, i1, ..., up, ip, up+1, ..., up+q, ip+q+1, ..., ip+q+r can uniquely be determined as a function of the other quantities. Hence if we terminate the first p ports with norators, the next q ports with current sources, the following r ports with voltage sources and the remaining ones with nullators then we obtain a uniquely solvable network.

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For example, one may terminate both ports of the first 2-port of Figure 2 with current 𝑢 sources, since 𝑢

𝑅

𝑅 𝑅

𝑅 𝑅

𝑅

𝑖 𝑖

holds and one can prove that any other

combination of voltage and current sources would do if none of the quantities R1+R3, R2+R3, R1R2+R1R3+R2R3 equals 0. The second 2-port is a voltage-controlled current source. Using voltage sources at both ports is possible, since

𝑖 𝑖

0 𝑔

0 0

𝑢 𝑢 , but

the only other way to obtain a uniquely solvable network is if we terminate the first port with a nullator and the second one with a norator, see Figure 3. The third 2-port of Figure 2 is a pair of a nullator and a norator (also called a nullor, see [Carlin and Youla]), its description (A|B) =

1 0 0 0

0 0 1 0

shows that it is only possible embedding into a

uniquely solvable network is if the first port is terminated by a nullator and the second one by a norator.

Figure 2: A 2-port with three resistors, a voltage-controlled current source and a pair of a nullator and a norator.

Figure 3: The two possible embedding of the voltage-controlled current source into a uniquely solvable network.

2 PRELIMINARY – PLANAR FRAMEWORKS Definition 2. One can consider a simple graph with n vertices and e edges as a planar bar-and-joint framework where the vertices are flexible joints and the edges are rigid bars (or rods).

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We have an intuitive feeling that a circuit of length 3 becomes a rigid framework while a circuit of length 4 does not unless we add an extra diagonal. Let (xi, yi) denote the coordinates of the location of the ith joint. If joints i and j are connected by a rod then their distance ((xi – xj)2 + (yi – yj)2)1/2 remains constant during any planar motion of the framework. Taking the time derivative of the square of this equation we obtain an equation of the form (xi – xj)(xi'– xj') + (yi – yj)(yi'– yj') = 0 for each rod. These equations together form a system Wz = 0. Here the elements of z are the quantities x1', x2', …, xn', y1', y2', …, yn', that is, the coordinates of the velocities of the joints. The rows of the matrix W of size e×2n correspond to the bars: If an edge connects vertices i and j then in its row the ith, jth, (n+i)th and (n+j)th elements are

xi – xj , xj – xi ,

yi – yj and yj – yi , respectively, while the other entries are 0. If the whole framework moves like a rigid body, without deformations, then the coordinates of the velocities of the joints satisfy this system of equations. Since all the congruent motions of the plane form a 3-dimensional subspace of the 2n-dimensional space, r(W)≤2n–3 always holds. The framework will be called (infinitesimally) rigid if r(W)=2n–3. For example, in case of the aforementioned framework (obtained from the circuit of length 4) the rigidity matrix W will be 𝑾

𝑥 –𝑥 0 0 𝑥 –𝑥

𝑥 –𝑥 𝑥 –𝑥 0 0

0 𝑥 –𝑥 𝑥 –𝑥 0

0 0 𝑥 –𝑥 𝑥 –𝑥

𝑦 –𝑦 0 0 𝑦 –𝑦

𝑦 –𝑦 𝑦 –𝑦 0 0

0 𝑦 –𝑦 𝑦 –𝑦 0

0 0 𝑦 –𝑦 𝑦 –𝑦

Its rank is at most 4 (that is, less than 2∙4 – 3), hence the framework cannot be rigid.

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3 AN ANALOGY BETWEEN ELECTRIC NETWORKS AND PLANAR FRAMEWORKS Both in case of electric networks and statics, we have a matrix – either (A|B) or W – with n pairs of columns related to each other, and even this formal analogy led to some interesting results, see [Recski, 1984a, b, c, d]. For example, if k=n for an n-port and the voltage columns of the matrix M=(A|B) are linearly independent (that is, if det A ≠ 0) then the network, obtained by terminating the ports by current sources (say, by open circuits prescribing i=0), is uniquely solvable. On the other hand, if all the columns of W except those, corresponding to {xi, yi, xj, yj, xk, yk,…} are linearly independent then the framework becomes infinitesimally rigid by fixing the joints i, j, k…. Transforming these basic, well-known results into the other theory, the analogy of the first sentence means that we keep the y-coordinates of the velocities of the joints zero, that is, we place the joints to horizontal „tracks” (which, of course, does not prevent a horizontal congruent motion even in case of rigid frameworks). On the other hand, the analogy of the second sentence means that we prescribe both the voltage and the current of the ports i, j, k… and ask if the resulting network becomes uniquely solvable (that is, we terminate these ports by nullators and the others by norators). A more complex example will be presented in Section 5. If there is some symmetry in network theory or statics, will this analogy lead to a meaningful symmetry in the other discipline?

4 SOME REFLECTION-TYPE SYMMETRIES The rigidity properties do not change if we reflect the planar framework along the 45° line x=y. Then, by analogy, we can interchange the roles of voltage and current. This symmetry is often called duality in electric network theory. However, there are two voltage-current symmetries, see [Iri and Recski], where we call the new n-port the inverse of the original one and reserve the concept dual for an n-port whose column space matroid is dual to that of the original one.

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For example, the 2-port described by u2 = cu1, i1 = 0 is a voltage-controlled voltage source. Its inverse is a current-controlled current source given by i2 = ci1, u1 = 0. However, its dual is another voltage-controlled voltage source with the role of the ports interchanged, that is, u1 = –cu2, i2 = 0. Using the analogy in the other way round, this inverse symmetry in electric network theory will correspond to the reflection symmetry of the planar framework. On the other hand, the duality symmetry in electric network theory will not correspond to a general symmetry among the frameworks – there are frameworks so that the dual of the column space matroid of W will not correspond to any framework. Using this analogy one may translate physical properties of electrical networks into those of planar frameworks but so far no interesting result has been found. For example, if the framework is a triangle, its 3×6 rigidity matrix can be considered as the description of a 3-port. One of the most fundamental electric properties of multiports is reciprocity but one can show that this 3-port will be reciprocal if and only if the three joints of the framework are collinear.

5 SOME ROTATION-TYPE SYMMETRIES The circulators are electric multiports with rotational symmetry. Let n ≥ 3. The n-port circulator is defined as a1=b2, a2=b3,… an=b1, where aj=uj+ij, bj=uj–ij for j=1, 2,…, n. The physical meaning of aj and bj is the incoming and the reflected wave at port j, respectively. For various properties of circulators see [Carlin and Youla], [Recski and Vékássy]. If n is even, the aforementioned analogy leads to the planar framework, corresponding to the circuit of length n. This will be illustrated for n=4, see also [Péterfalvi and Recski]. Recall that the rigidity matrix is

SYMMETRIES IN ELECTRIC NETWORKS

𝑾

𝑥 –𝑥 0 0 𝑥 –𝑥

𝑥 –𝑥 𝑥 –𝑥 0 0

0 𝑥 –𝑥 𝑥 –𝑥 0

0 0 𝑥 –𝑥 𝑥 –𝑥

𝑦 –𝑦 0 0 𝑦 –𝑦

𝑦 –𝑦 𝑦 –𝑦 0 0

101

0 𝑦 –𝑦 𝑦 –𝑦 0

0 0 𝑦 –𝑦 𝑦 –𝑦

.

Suppose that the positions of the joints are generic. Then the rank of W is 4, hence we should delete 4 columns to obtain a non-singular 4×4 matrix. (a) If we delete columns 1, 3, 5, 7 then the remaining matrix will be non-singular – this corresponds to case (a) of Figure 4 where we fix the positions of joints 1 and 3. (b) However, if we delete columns 1, 2, 5, 6, that is, we fix the positions of joints 1 and 2 then the remaining matrix will be singular, the framework as a whole will have a motion. (c) Deleting columns 1, 2, 5, 7 means that joint 1 is fixed, joints 2 and 3 are placed to a vertical and a horizontal track, respectively, see case (c) of the figure. In this way, the position of every joint is fixed and the matrix will be non-singular. (d) Deleting columns 1, 2, 3, 8 means that joints 1, 2 and 3 are placed to vertical tracks while joint 4 to a horizontal one. Again, in this way, the position of every joint is fixed and the matrix will be non-singular. (e) However, deleting columns 1, 2, 3 and 4 means that every joint is on vertical tracks. Then the remaining matrix will be singular, the framework as a whole will have a motion.

Figure 4: The framework becomes rigid in cases (a), (c) and (d) but remains nonrigid in cases (b) and (e).

Using the aforementioned analogy one can conclude that the networks of Figure 5 are uniquely solvable: the first one corresponds to Figure 4(a) since nullators and norators correspond to fixed and free joints, respectively, and the second one to Figure 4(c) since voltage and current sources correspond to vertical and to horizontal tracks, respectively.

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The column space matroid of W (in the generic case) is a rank 4 matroid on an eight element set. It has a transparent representation in the three-dimensional affine space, see Figure 6. There are two types of minimal dependent sets, that is, coplanar four-tuples. Those lying on the planes P1, P2 correspond to motions like the one shown on Figure 4(e), the others, lying on two intersecting lines correspond to motions like the one shown on Figure 4(b).

Figure 5: The circulator networks corresponding to Figures 4(a) and 4(c).

Figure 6: The affine representation of the column space matroid of the rigidity matrix of a generic planar circuit of length four.

The same correspondence works for any even n > 4 as well. For example, there is a third type of minimal dependency for n = 6, the corresponding motion is shown in Figure 7.

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However, this analogy does not work if n is odd. For example, the column space matroid of the describing matrix of the 3-port circulator is isomorphic to the cycle matroid of the complete graph with 4 vertices, see [Iri and Recski]. Hence the three voltage columns are linearly dependent but the three current columns are linearly independent. On the other hand, both the first n columns and the last n columns of a rigidity matrix must be linearly dependent.

Figure 7: The “third type” of motion of a generic planar circuit of length 6.

Acknowledgement: The research reported in this paper was supported by the BMEArtificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC) and by the grants OTKA #124171 and BME NC TKP2020 of NKFIH Hungary.

REFERENCES Carlin, H. J., and Youla, D. C. (1961) Network Synthesis with Negative Resistors, Proc. IRE, 49 907-920. https://doi.org/10.1109/JRPROC.1961.287934 Iri, M., and Recski, A. (1980) What does duality really mean? Internat. J. Circuit Th. Appl. 8 317-324. https://doi.org/10.1002/cta.4490080311 Péterfalvi, F., and Recski, A. (2020) Egy villamosságtani probléma, matroidelméleti megoldással és statikai következménnyel [A problem in electric network theory, with a matroidal solution and with a corollary in statics, in Hungarian] Alkalmazott Matematikai Lapok, https://doi.org/10.37070/AML.2020.37.2.12 Recski, A. (1984a) A network theory approach to the rigidity of skeletal structures I. Modelling and interconnection, Discrete Applied Mathematics 7 313-324. https://doi.org/10.1016/0166218X(84)90007-6 Recski, A. (1984b) A network theory approach to the rigidity of skeletal structures II. Laman’s theorem and topological formulas, Discrete Applied Mathematics 8 63-68. https://doi.org/10.1016/0166218X(84)90079-9 Recski, A. (1984c) A network theory approach to the rigidity of skeletal structures III. An electric model for planar frameworks, Structural Topology 9 59-71

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Recski, A. (1984d) Statics and electric network theory: a unifying role of matroids, in W. R. Pulleyblank ed. Progress in combinatorial optimization, London, Academic Press, 307-314. https://doi.org/10.1016/B978-0-12-566780-7.50024-6 Recski, A., and Vékássy, Á. (2020) Can the genericity assumption decrease the rank of a matrix? Periodica Polytechnica Electrical Engineering and Computer Science. https://doi.org/10.3311/PPee.16647

Discrete Lattices on the Single Bearing Spiral: From Geometry to Botany Dmitriy Gurevich

Geologist, Consultant Almaty, Kazakhstan Email: [email protected]

Abstract: This work is based on the “Bearing Spiral Hypothesis”. It refers primarily to organic objects with helical or spiral structures and shapes. It assumes the existence of a single bearing (generative) spiral (or helix) that sets the trajectory of the elements’ multiplication. The elements, regularly located on the bearing spiral, form discrete lattices that can be studied using elementary geometry. The key point of the hypothesis is the interpretation of the “mode of existence” of the bearing spiral. Since nothing has been identified resembling this spiral “in physical expression” in plant anatomy, it is necessary to initially agree that the bearing spiral can be a spatial expression of a cyclic process.

1 INTRODUCTION The shape and the structure of a real object depend on external and internal factors, on the interaction of its own symmetry and the symmetry of the environment. So, the shape of a particular crystal is determined by the type of crystal lattice and by the conditions in which it grew and existed. Likewise, the shape of a particular organism is determined by the blueprint encoded in the DNA and the conditions in which that design was implemented. The most important difference is that the ”embryo“ of a crystal – a kind of a ”seed“ on which crystallization begins – can be brought in from the outside, or it can be formed ”here and now“; on the contrary, the “germs” of organisms, their DNA, are always “seeded”. The most important similarity is the determining role of symmetries in both cases. The plan of the organism, rational and effective ‘by default’,

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cannot but uses elements and operations of symmetry - the most effective tools for the information folding and unfolding. This is what we observe in ourselves (mirror leftright symmetry, symmetry-antisymmetry of the top and bottom, the similarity of the structure of the hands and the whole body, etc.). We observe this around us – in the forms of other people and animals. Some elements of symmetry are striking and not questionable. Others require special attention – some of them can be distorted under the external influences, some of them are too familiar and do not cause interest, and some of them, on the contrary, are unfamiliar and therefore invisible. All this fully applies, for example, to inflorescences of plants and shells of molluscs, which have complex helical and spiral symmetries. Concerning botany, biology and geometry, the phenomenon of spiral symmetry has been studied by many researchers (Church, 1904; Cook, 1914; Thompson, 1942; Coxeter, 1969; Jean, 2009; Sadoc et al., 2012, for a detailed review see Adler et al., 1997). The largest number of works is devoted to plants. To describe them, the concept of phyllotaxis was formulated that characterizes the arrangement of leaves on the stem (often spiral), flowers and seeds in fruits and inflorescences (usually spiral). Particular interest is in this topic because in many cases phyllotaxis follows the golden ratio with very high accuracy. While early works on this topic were focused on the geometric descriptions, the emphasis has shifted towards the study of the molecular mechanisms of phyllotaxis in recent decades (Gola et al., 2016; Okabe et al, 2019, etc.). This is in a full agreement with the main trend in natural science development. However, in the author's opinion, there are still many blank spots in the study of forms and structures. This also applies to the possibility of identifying not yet studied “geometric species”, and the possibility of their original classification. This, in turn, can significantly help in the creation of a theory. For example, we can take the Bravais lattices and Fedorov symmetry groups – purely geometric constructions that had formed the basis of crystallography and crystallochemy. Moreover, the tradition of considering botanic patterns as geometric lattices originates in the work of Auguste Bravais and his brother Louis (Bravais, 1839). The tradition of identifying such a spiral as a “supporting structure” originates in the works of Alexander Braun and Karl Schimper (1836), who were the first to single out a spiral structure in genetics, also known as the generative, fundamental and ontogenic one (Jean, 2009).

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In the next section, we will briefly but, if possible, strictly describe the concepts that will then be used in describing the spiral lattices. Further, a classification table will be built and its most general patterns and groupings will be considered. Then we will turn to some of the prominent examples of spiral structures such as the “Fibonacci lattices”. The second part of the work (Part 2: From botany to music and back), which was published in Symmetry: Culture and Science (Gurevich, 2020), filled the selected cells of the classification table, that is, described examples of spiral lattices implemented in nature and culture.

2 DEFINITIONS Both helical and spiral symmetries are composed of elementary symmetry transformations. As in the classical crystallography, helical symmetry is constructed from translation along an axis and rotation (rotational transformation) around it (Fig. 1.a). Spiral symmetry is limited to one plane and is built from translation from the centre and rotation around it (Fig. 1.b). The first “organic peculiarity” lies in the fact that the ratios of the three listed parameters can change – both depending on the place and the stage of development. For example, a growing sunflower progresses from a “stem stage” to a “basket stage” (Fig. 1. a, b). At the “construction” of the pineapple fruit, the vertical and horizontal translations are constantly connected by a certain non-linear function (Fig. 1.c). Another feature is that the helical and spiral lines formed by the above symmetries become the trajectories of additional translations and the locations of the growth points, and thus determine not only the shape but also the structure of the object.

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Figure 1: four variants of the “bearing spiral” formation. a. Interaction of perpendicular linear translation along a and rotation  (helix), b = const (“stem”). b and c. Interaction of linear translation in the plane along b and rotation , a = 0, convergent (b) and divergent (c) cases (“sunflower basket”, spiral). d. Interaction of all three parameters (“pineapple” helix).

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The paper investigates spiral structures that are combinations of radial translation (from the centre), main rotation around this centre and additional translations. The radial translation and the main rotation form a spiral, which we will further call the ”bearing” one (Fig. 1.b, c). This spiral defines the trajectory of the additional translation which ”multiplies“ the elements and forms the spiral lattice1. The nature of the lattice, therefore, depends on the type of the bearing spiral and the magnitude of an additional translation, determined in relation to one coil of the bearing spiral. If the angular size of the translation is 2π, then there is one element per turn, if it is equal to π – two elements, etc. Resulting structures can be considered as infinite discrete lattices with finite elementary cells of the [m/n] type, where m and n are natural numbers; m is the number of convolutions in the elementary cell, and n is the number of translations in it (Fig. 2.a). If one coil contains an integer number of translations, then the elementary cell corresponds to [1/n]. Besides, n determines the total number of sectors in the elementary cell, and m – the number of sectors per elementary translation . The angular size of generative translation or angle of divergence  can be calculated from the equation  = 2π*m/n, and sectorial angle σ – from the equation σ = 2π/n. We will call such lattices or structures, in short, “single-spiral lattices” or “spiral structures” or “[m/n] lattices”. We will consider mostly the simple lattices formed by a single sequence of translations. If we deal with several sequences, we will use the notations 1, 2, 3… for the divergence angles and 1, 2, 3… for translations. There are some terms below, important for the following narration (based on the Glossary by Jean, 1994). Phyllotaxis – arrangement of the leaves around the stalk. Divergence angle (or simply the divergence) – the angle between the lines drown from the centre to any two successive elements. Ortostich and parastich – correspondently linear (radial) and spiral lines, observed in the structure of plants (stems with leaves, fruits, cross-sections of apexes, etc.). Family (or series) of parastiches – a set of parastiches with the same pitch, winding around a common pole in the same direction. We will use several not common designations. Family (or series) of ortostiches means the total number of ortostiches (radial lines). We will designate ortostiches and parastiches as O and P correspondingly. The density of the [m/n] lattice is equal to the

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The definition was taken from (Rothen F. and Koch A.-J., 1989).

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Figure 2: Discrete lattice of [m/n] type formed by the regular translations ( = const) along the single bearing spiral. Left: a geometric representation of the [5/13] lattice, above – centric (term by Jean, 1994), below – cylindrical representations; the elementary cell is shadowed. Right: the ’natural examples’ of the [5/13] lattice, above - fir cone, bottom view, below – cedar cone, side view. H – the height of the elementary cell (5 coils), h – coil’s height, M – one coil of the generative spiral.

Figure 3: Definitions used for the directions of translations and growth of the generative spirals and their „enantiomorphic signs”. Upper left square denotes the default situation.

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arithmetical value of the m/n fraction. By default, we consider divergent (“untwisted”) translations and the generative spiral growing clockwise (upper left square on Fig. 3).

3 THE CLASSIFICATION TABLE Table 1 shows a fragment of the classification matrix; Figure 4 explains notations. Each [m/n] lattice is characterized by a particular pattern, formed by the elements (represented by the black dots), translated along single bearing spirals and by the parastiches or secondary spirals, represented by thick lines joining the neighbouring elements2.

Figure 4: Notations used in the classification table (see: Table 1).

The First group3 combines the simplest structures, which cells are coloured blue (grey in black/white print). The most primitives of them belong to the ”main diagonal“, they have one translation per one turn and correspond to the cases m = n (line 1). Other simple structures correspond to the simple fractions with n and m = 1, 2, and 3; or to the fractions which can be arithmetically reduced to the simple ones ([2/6], [6/8]…), or fractions which could be rewritten as the simple fraction plus integer ([7/2] = [1/2] + 6…). Lattices, related to the fractions that can be reduced (such as [2/6]) are identical to those, related to the reduced fractions ([1/3]). All these structures have no parastiches, correspondingly they look similar to the structures formed by the classical rotational symmetry.

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Smooth connection of elements (more precisely, their centres) is the simplest way to image the parastiches. In fact, as seen in Figure 2, parastiches can be composed of the elements’ boundaries. This issue is discussed in more detail below. 3 The concept of a "group" is used here and below in the sense of a complex of lattices that have certain general properties, and not in the strict sense of a concept from set theory.

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The second group includes fairly simple structures that have only single families of parastiches, and the number of parastiches in the family does not exceed three. Their cells are coloured yellow (light grey). Single-tailed structures, possessing only one secondary spiral, correspond to the cases when the difference of n and m is equal or close to 1. If m > n, the secondary spiral rotates clockwise (in the same direction as the generative spiral), if m < n it rotates in the opposite direction. We can note a sequence of “one-tailed“ lattices, wich starts with a certain [m/n] and continues as [(m+n*N)/n], where N is an integer. For example: [6/7] - [13/7] - [20/7] ... . The “two-tailed” structures, with one family of two parastiches, correspond to the fractions in which n is equal to or greater than 7, n/m = 2 ± 1, and n/(n-m) = 2 ± 1 ([4/9], [5/9]…). If 2m> n – the parastiches are twisted to the left (“+”), if 2m