Polytopes and Symmetry 0521277396, 9780521277396

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor:

Professor J.W.S.

Cassels,

Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane,

Cambridge CB2

1SB.

I.

General cohomology theory and K-theory,

4.

Algebraic topology,

P.HILTON

J.F.ADAMS

5.

Commutative algebra,

8.

Integration and harmonic analysis on compact groups,

J.T.KNIGHT

9.

Elliptic

functions and elliptic

10.

Numerical

II.

New developments

ranges II,

12.

Symposium on complex analysis, & W.K.HAYMAN

13.

R.E.EDWARDS

P.DU VAL

F.F.BONSALL & J.DUNCAN

in topology,

G.SEGAL

(ed.)

Canterbury,

1973,

J.CLUNIE

(eds.)

Combinatorics: 1973,

curves,

Proceedings of the British Combinatorial Conference

T.P.MCDONOUGH & V.C.MAVRON

introduction to topological

(eds.)

15.

An

16.

Topics

17.

Differential

18.

A geometric approach to homology theory,

in finite

groups,

germs

groups,

P.J.HIGGINS

T.M.GAGEN

and catastrophes,

Th.BROCKER & L.LANDER S.BUONCRISTIANO,

C.P.

ROURKE

& B.J.SANDERSON 20.

Sheaf

21.

Automatic continuity of linear operators,

theory,

B.R.TENNISON

23.

Parallelisms of complete designs,

24.

The

topology of Stiefel manifolds,

25.

Lie

groups

26.

Transformation groups:

and compact groups,

I.M.JAMES

J.F.PRICE

Proceedings of

of Newcastle-upon-Tyne,

August

27.

Skew field constructions,

28.

Brownian motion.

29.

Pontryagin duality and the

A.M.SINCLAIR

P.J.CAMERON

1976,

the conference

in

the University

C.KOSNIOWSKI

P.M.COHN

Hardy spaces and bounded mean oscillations,

K.E.PETERSEN groups,

structure of locally compact Abelian

S.A.MORRIS

30.

Interaction models,

31.

Continuous

N.L.BIGGS

crossed products and type III von Neumann algebras,

A.VAN DAELE 32.

Uniform algebras and Jensen measures,

33.

Permutation groups

34.

Representation theory of Lie

35.

Trace

36.

Homological

37.

Partially ordered rings and semi-algebraic geometry,

38.

Surveys

39.

Affine

40.

Introduction to Hp spaces,

41.

and combinatorial groups,

ideals and their applications, group theory,

in combinatorics, sets

C.T.C.WALL B.BOLLOBAS

and affine groups,

T.W.GAMELIN structures, M.F.

N.L.BIGGS & A.T.WHITE

ATIYAH et al.

B.SIMON (ed.) G.W.BRUMFIEL

(ed.)

D.G.NORTHCOTT

P.J.KOOSIS

Theory and applications of Hopf bifurcation,

B.D.HASSARD,

N.D.KAZARINOFF & Y-H.WAN 42.

Topics

43.

Graphs,

in the codes

theory of group presentations, and designs,

44.

Z/2-homotopy theory,

45.

Recursion theory: &

S.S.WAINER

D.L.JOHNSON

P.J.CAMERON & J.H.VAN LINT

M.C.CRABB

its generalisations and applications,

F.R.DRAKE

(eds.)

46.

p-adic analysis:

47.

Coding the Universe,

a

short course on recent work,

48.

Low-dimensional

A.BELLER,

topology,

R.JENSEN &

N.KOBLITZ

P.WELCH

R.BROWN & T.L.THICKSTUN

(eds.)

49.

Finite

geometries and designs,

& D.R.HUGHES

P.CAMERON,

J.W.P.HIRSCHFELD

(eds.)

50.

Commutator calculus and groups

51.

Synthetic differential

of homotopy classes,

52.

Combinatorics,

53.

Singularity theory, V.I.ARNOLD

geometry,

H.N.V.TEMPERLEY

(ed.)

54.

Markov processes and related problems of analysis,

55.

Ordered permutation groups,

56.

Journees arithmetiques

57.

Techniques of geometric Singularities of

59.

Applicable differential

60.

Integrable

61.

The

J.V.ARMITAGE

topology,

(ed.)

R.A.FENN

smooth functions and maps,

systems,

core model,

geometry,

J.MARTINET

M.CRAMPIN & F.A.E.PIRANI

S.P.NOVIKOV et al.

A.DODD

62.

Economics

63.

Continuous

64.

Basic

65.

Several

complex variables and complex manifolds I,

66.

Several

complex variables and complex manifolds II,

67.

Classification problems

68.

Complex algebraic surfaces,

69.

Representation theory,

for mathematicians, semigroups

J.W.S.CASSELS

in Banach algebras,

A.M.SINCLAIR

concepts of enriched category theory,

70.

Stochastic differential

71.

Groups - St Andrews

72.

Commutative algebra:

73.

E.B.DYNKIN

A.M.W.GLASS

1980,

58.

H.J.BAUES

A.KOCK

Riemann surfaces:

in ergodic theory,

G.M.KELLY M.J.FIELD M.J.FIELD

W.PARRY & S.TUNCEL

A.BEAUVILLE

I.M.GELFAND et al. equations on manifolds,

1981,

K.D.ELWORTHY

C.M.CAMPBELL & E.F.ROBERTSON

Durham

1981,

R.Y.SHARP

a view towards

several

an algebraic

approach,

(eds.)

(ed.)

complex variables,

A.T.HUCKLEBERRY 74.

Symmetric designs:

75.

New geometric splittings of classical knots

E.S.LANDER (algebraic knots),

L.SIEBENMANN & F.BONAHON 76.

Linear differential operators,

77.

Isolated singular points on complete

78.

A primer on Riemann surfaces,

79.

Probability,

80.

Introduction to the representation theory of

82.

Skew fields, Surveys

84.

Finite

J.F.C.KINGMAN & G.E.H.REUTER compact and

(eds.)

locally

A.ROBERT

in combinatorics:

Homogeneous

E.J.N.LOOIJENGA

P.K.DRAXL

Combinatorial Conference 83.

intersections,

A.F.BEARDON

statistics and analysis,

compact groups, 81.

H.O.CORDES

Invited papers 1983,

E.K.LLOYD

for the ninth British (ed.)

structures on Riemannian manifolds,

group algebras and their modules,

85.

Solitons,

86.

Topological

87.

Surveys

88.

FPF ring theory,

F.TRICERRI & L.VANHECKE

P.LANDROCK

P.G.DRAZIN topics,

I.M.JAMES

in set theory,

(ed.)

A.R.D.MATHIAS

(ed.)

C.FAITH & S.PAGE

89.

An F-space

90.

Polytopes and symmetry,

sampler,

N.J.KALTON,

N.T.PECK & J.W.ROBERTS

S.A.ROBERTSON

London Mathematical Society Lecture Note Series:

90

//

Polytopes and Symmetry STEWART A.

ROBERTSON

Professor of Pure Mathematics, University of Southampton

CAMBRIDGE UNIVERSITY PRESS Cambridge London

New York

Melbourne

Sydney

New Rochelle

Trent University U&rery

PeTEfieOROUQK, out.

Published by the Press The Pitt Building,

Syndicate

32 East 57th Street,

New York,

296 Beaconsfield Parade,

NY

Cambridge CB2

10022,

Middle Park,

©Cambridge University Press First published

of the University of Cambridge

Trumpington Street,

Melbourne 3206,

1984

Library of Congress

catalogue card number:

Stewart A.

Polytopes and symmetry Society lecture note Polytopes

I.

Title

512'. 33

ISBN

0

521

II.

Series

QA691

27739 6

(London Mathematical

series,

Cambridge

83-15171

British Library Cataloguing in Publication Data

1.

Australia

1984

Printed in Great Britain at the University Press,

Robertson,

1RP

USA

ISSN 0076-0552;

90)

Contents Preface

vii

Synopsis 1.

2.

3.

The Space of Polytopes 1.

Euclidean space

1

2.

Affine hulls of finite sets

2

3.

Lattice structure

3

4.

Polytopes

3

5.

The Hausdorff metric

4

6.

The space of polytopes

6

7.

Similarity and congruence

8

8.

Euclidean similarity

9

9.

The similarity space of polytopes

11

Combinatorial structure 1.

Facial structure

12

2.

Combinatorial equivalence

14

3.

Topological structure of combinatorial types

18

4.

The three standard sequences

23

5.

Simple and simplicial polytopes

24

6.

Duality and polarity

25

7.

Joins

28

8.

Cones

29

9.

Regularity

30

Symmetry equivalence 1.

Transformation groups

35

2.

Slices

36

3.

Normal polytopes

39

4.

Symmetry equivalence

40

5.

Symmetry types as orbit types

42

6.

Symmetry invariants

44

7.

Symmetry equivalence and polarity

46

VI

4.

Products and sums

.

5.

1

Linear decomposition

47

2.

The rectangular product

50

3.

Combinatorial structure

51

4.

Lattice products

51

5.

Face-lattice of rectangular products

52

6.

Combinatorial automorphisms

53

7.

Symmetry groups

54

8.

Deficiency and perfection

55

9.

The rectangular sum

56

Polygons

.

6.

1

Combinatorial structure

58

2.

Possible symmetry groups

60

3.

Symmetry types

61

4.

Deficiency

62

5.

Triangle s

64

6.

Quadrilaterals

68

7.

The

71

1-skeleton

Polyhedra

.

1

Some combinatorial properties of polyhedra

74

2.

Cuboids

75

3.

Vertex-regular polyhedra

79

4.

The nonprismatic groups

84

5.

The prismatic groups

93

6.

Face-regular polyhedra

96

7.

Edge-regular polyhedra

97

8.

Perfect polyhedra

100

Concluding remarks

101

Bibliography

104

Index of symbols

107

Index of names

109

General Index

110

Preface These notes are intended to give a fairly systematic exposition of an approach to the symmetry classification of convex polytopes that casts some fresh light on classical ideas and generates a number of new theorems.

The theory is far from complete, and there is a range of

attractive unsolved problems.

I hope that the level of sophistication

will he found suitable for anyone with a knowledge of contemporary pure mathematics at first degree level. The work has developed sporadically over almost twenty years. In the early stages,

I worked in collaboration with Sheila Carter and

Hugh Morton at Liverpool.

At that time we concentrated on highly

symmetric polytopes, but many of the key ideas apply equally well to the study of polytopes in general.

We were greatly helped in these days by

the generous advice and encouragement of C.T.C. Wall.

Morton's

influence is particularly strong in Chapter 4, which is based on previously unpublished joint work, while Chapter 6 owes much to Sheila Carter's patient investigation of symmetric polyhedra. Southampton in 1970, geometrical insight.

Since coming to

I have received the benefit of A.W. Deicke's acute Deicke’s contributions to my understanding of

cuboids and polyhedra generally have been considerable.

Other

colleagues, notably David Chillingworth, Gareth Jones and David Singerman, have helped me by producing good examples and neat lines of argument in response to my questions.

In recent years, D.G. Kendall has

been developing a theory of shapes for finite sets of points in Euclidean space that has something in common with my approach to polytopes, although Kendall is concerned with a rather different class of problems. His concern is with statistical questions, and convexity does not hold the central position that I have given it here.

Kendall's work makes

considerable use of metrical ideas, which I have used only to generate topologies.

Nevertheless,

there is a common spirit in the two theories,

and the reader who sees anything of interest in this account will certainly derive great benefit from a study of Kendall's work. Kendall

(1983)

See

for a recent version and references to previous and forth¬

coming articles. Another substantial piece of work that is related to my approach to polytopes is Schwarzenberger's study of crystallography in n-dimensional space

(Schwarzenberger (1980)).

viii

Schwarzenberger gives a systematic treatment of the concepts needed to provide a coherent framework for the classification of 'crystal'

structures in Euclidean n-space

tures involved are lattices in

En

En

.

The geometrical struc¬

rather than polytopes,

and the groups

are infinite discrete subgroups of the Euclidean group rather than finite subgroups of the orthogonal group.

The situation in crystal¬

lography is more difficult to handle than in the theory of polytopes, since the symmetry groups themselves admit continuous deformations. I thank I.M. James for his patient encouragement, June Kerry for her efficient work in preparing the typescript,

and D.G. Kendall for

his friendly attitude in giving me access to some of his unpublished work.

Southampton June,

1983.

S.A. Robertson

0.

SYNOPSIS

The following is a brief outline of some of the main ideas and results that are discussed in detail in the main body of the text. There is no direct relation between the organisation of the paragraphs here and the arrangement of the various chapters. 1.

Euclidean space The polytopes discussed in this book are convex polytopes in

Euclidean n-space

E

,

where

n

is any positive integer.

So that all

polytopes can be treated simultaneously as members of a single family, it is convenient to regard

En

as a fixed linear subspace of

Consequently, we employ the union

E

of all the spaces

receptacle for all polytopes, referring to 2.

E

En

En+1

.

as a

as Euclidean space.

Polytopes Our objects of study are the convex hulls

finite subsets conv A

A

of

E

.

The dimension

dim P

is the dimension of its affine hull

called an n-polytope if

dim P = n

.

P = conv A

of the

of a polytope

aff P = aff A ,

Thus the empty set

0

P =

and

P

is the

unique (-1)-polytope, while 0-polytopes are singleton subsets of 1-polytopes are closed bounded straight line segments in are

(convex,

3.

Symmetry

plane) polygons,

and 3-polytopes are

E

is

E ,

, 2-polytope

(convex) polyhedra.

The broad aim of the theory is the classification of polytopes according to their geometrical symmetry.

The first step in the creation

of such a theory is the choice of a basic equivalence relation on the family

, 1

The quotient space

consists of the O-dimensional symmetry types of all squares and of all 1-polytopes.

cm Squares *

rect«w«les

and

c l-polijte>pes

xii

8.

Deficiency For each polytope

action

ap

P

we try to find relations between the

and the topological invariants of the symmetry type

to which (the similarity class of)

P

belongs.

,'" n .

z e E

,

In this way

we obtain a sequence of inclusions -,n+l

and we can write

E =

En = lim En . n=l

The space

E

has a real linear structure given by

^(xi)

where

z^ = Ax^ + uy^

Y = (y-)

£ E

.

,

+ y(yi)

i = 1,...

= (zi)

, n,...

The metrical structure of

,

,

for

A,yeR ,

x = (x^)

E

is determined by the

,

0°

familiar Euclidean inner product for all

x,y c E

are then given by

.

< , >

The associated norm ||x |[

= /

and

The resulting metric space E

,

is called Euclidean space.

cause us no trouble, lie in some

En .

,

where ||

||

d(x,y) (E,d)

,

The fact that

x^y^

and Euclidean metric = ||x - y ||

respectively.

E

is not complete will

since we are interested only in subsets of

En

at once.

d

which we denote simply by

Our only reason for introducing

handle all the spaces

=

E

E

that

is to be able to

2

2.

Affine hulls of finite sets Le t

!}-

denote the family of all finite subsets of

denote the number of elements in

A e ^ by frA

iff

A

If

is a singleton

A e

,

{a}

,

a z E

.

We

(and extend this

notation to finite sets in general in Chapter 2). ^A = 1

E

Thus 4^0 = 0

and

.

then any linear combination

E

^ caa

f°r which

I

.t =1 is called an affine combination of A . The set of all aeA aaffine combinations of A is called the affine hul1 aff A of A , any such subset by

A .

X = affA

More generally,

of

affA,

A

A e ^ .

affA = A

iff

happen that

affA ,

then

Then A = 0

aff:(?->• A or

A

affA = affB

A

basis for

S

where

E

,

A z ^

affS

and

denotes the

ACS

.

is a surjective map.

is a singleton

{a}

for distinct sets S

of

A ,

A,

aff S

.

For example,

Of course it may

B e ^ .

If

affA = X

is a proper subset of

is said to be affinely independent and to be an affine

X. The set

is a subset of

A

*£>

of all finite-dimensional linear subspaces of

with inclusion

say.

give

0

-1 .

There is then a parallel projection

where

of

denote the set of all affine planes of the form

and for every proper subset X ,

is called an affine plane generated

for any subset

union of the affine planes Le t

E

and

'honorary membership' of

L = {x-y:x,y e X}

X, Y e ^

.

p:^->^>

poi = 1^ .

X || Y ,

iff

given by

p (X)

= L

,

We say that

P (X) CL P (Y)

or

.

The dimension

dim L

of any

extend this notion to the elements of for each

For convenience, we

assigning to it the dimension

Trivially,

are paral lei, written

F(Y) C PCX)

,

E

X z A •

We say that

L e A

is well-defined, and we

by putting

dim X = dim P(X)

X e A

is an affine k-plane

X = affA .

Then j^A 5 1 + dimX

iff

dim X = k . Exercise: equality Exercise:

Let iff

A e ^ and A

Let

Thus if ^

'^ = Un5l^n •

is affinely independent.

A

.

and j4n

affine planes in

with

En ,

Then

ACED

for some

n .

Hence

af f A C En .

denote the sets of all finite subsets and of all respectively,

then

y =unil

n

and

3

3.

Lattice structure Both

the set

2

3"

\A

and

inherit the partial order by inclusion from

of all subsets of

to this partial order. lower bound of

E ,

In 3 ,

A,B e 3

are

and both are lattices with respect

the least upper bound and greatest

A C) B

and

we denote the least upper bound of greatest lower bound of X U Y = X y \

iff

X, Y e jA

X C Y

or

Let

A,B e 3

af f (A U B) = X y Y ,

and

and

and

Y

is just

Y3 X ,

of all affine planes that contain Exercise:

X

A 0 B

respectively. by

X t)Y ,

XOY .

since

In •SA' #

while the

The fact is that

X y Y

is the intersection

X U Y .

X = affA,

Y = affB .

aff(AC)B)CI X O Y .

Then

F ind

A C B =>

A, B

XCY ,

such that

the last inclusion is proper. Exercise: p(X W Y)

4.

For all

L,M

= p(X) +p(Y)

L fe) M = L + M . and

For all

p (X H Y) C P (X) A P 00

.

Polytopes Let

all

aeA,

A £ 3 • t_ 5 0 3.

.t a aeA a is called a convex combination of ““

of

wThere

Any subset

P = conv A ,

polytope or simply a polytope.

1

- •

A

A .

Z

Any affine combination

set of all convex combinations of

■

■

dim(affA)

,

e

is said to be of dimension

and we refer to

The empty set

A ,

. ——

A e 3 >

A Cl conv A Cl affA = aff(convA)

P = convA

such that, for

is called the convex hull

It follows at once from the definitions

If

X,Y e vsA ,

The

1-polytopes are closed

2-polytope is called a polygon and a

3—polytope is called a polyhedron. Notice that if

A, B £ 3

A more subtle fact is that if conv(A set

B)

V e 3

conv W E

D

.

anci

A C B ,

conv A = conv B = P ,

It follows that for each polytope

such that

P = conv V

is a proper subset of

whose convex hull is

P

.

then conv A Cl conv B .

P .

P

then

P =

there is a unique

and for every proper subset Thus

V

The elements of

is the V

W

of

V ,

'smallest' subset of

are called the vertices

of

P ,

and

V

is called the vertex set

tionship between

vert

and

conv

(P

-*

vert P

of

P

The rela-

.

is shown in the commutative diagram

V Thus

vert

is injective and

conv

is surjective.

It follows from the first Exercise of §2 that an n-polytope has at least vert P

n + 1

vertices,

and has exactly

is affinely independent.

that an n-polytope

P

simp1ex.

E .

n + 1

iff

vertices

iff

these vertices

Such an n-polytope is called an n-

-1 £ n $ 1 ,

Of course for

vertices

Another way of putting this is to say

has exactly

are 'in general position' in

n + 1

P

every n-polytope is an n-simplex.

A 2-simplex is called a triangle and a 3-simplex is called a tetrahedron for reasons that either are familiar already or will become clear shortly. With the second Exercise of § 2 in mind, we note that for each

r

y

A:F(P) -> F(Q) determined

A(A.) = B for some labelling of F _^(Q) Now J 1 = aff B. is given by a unique equation = 1 ,

K.

is sufficiently small, where

Let

£•

= w.

- v.

and

111

b. I 0 J T b. - a. Since J J

n• J

= 1

and

i

Let the combinatorial equivalence

B be such that

En

in the sense that its vertices are

B:FQ(P) - Fq(Q)

isomorphism.

provided

P

where, for some

and the map

by

to

Q

Uij(P)

=

j(Q)

for all

= 1

iff

(P) = 1

,

and hence

+ +