Idiot's Guides: Geometric Origami Kit 9781615648481, 1615648488

Citation preview

Geometric Origami Kit by Nick Robinson

A member of Penguin Random House LLC

Introduction The world of geometry has aspects that resonate throughout the natural world—you only need to look at fractals, logarithmic spirals, basalt rocks at the Giant’s Causeway, and even the shapes of honeycombs within a beehive. All of this can inspire creativity within origami. In a chaotic world, geometry offers order, symmetry, and beauty. This has been the case since the days of Plato and probably even earlier. Geometry lies at the very heart of origami, regardless of what you make. All you need to do is to unfold an origami model and study the revealed crease pattern. Within it, you will find triangles, squares, and many other polygons, often revealing strong symmetry. To many people, the deeper you look into it, the more fascinating it becomes. There is also an interesting dichotomy between the impermanence of paper and the apparently solid shapes you can construct using it. Whether you are new to origami or have experience with it, this book provides practical possibilities for creating geometric designs. By following the designs in the order they are presented, you can build on your base of knowledge and produce beautiful, finished results. This book may even inspire you to do some research to understand the angles and ratios that make up each design! Fold slowly and neatly, and enjoy the exciting designs in this book!

Acknowledgments I would like to thank the following creators for kindly allowing their work to be used within this book: Robert Foord (Diamonds), Mick Guy (Versatile), Hans-Werner Guth (Eifel Star/Jump Unit), Larry Hart (Snap Hexahedron), Mitsunobu Sonobe (Sonobe Unit), Boaz Shuval (Dodecahedron), Dave Brill (Segmented Unit), Vignesh Cumareshan (Bow Tie Cube), Francesco Guarnieri (Diamond Star), Francis Ow (Truncated Star), Denver Lawson (Bipyramid Blackberry/Bipyramid Dragon Fruit/Leaved Octahedral Skeleton), and various (Spinning Top/Truncated Cube). With any origami design, there is a reasonable chance that similar designs have been independently created by others. Therefore, I’d also like to give credit to Tomoko Fuse and Kunihiko Kasahara, both creative giants of geometric origami.

Thanks to Susan Zingraf, Kayla Dugger, William Thomas, and Mike Sanders at Alpha Books for the support and professionalism. To Tom Cuffe (of Irish Origami fame), thanks for proofreading the diagrams. To the British Origami Society, thank you for the support and inspiration. I’d also like to thank my many origami friends around the world for their boundless friendship. To Rachel and the Riffs, thanks for the musical fun. Thanks to Andy Peake, the other half of my group Lost Garden. Last (but not least), I’d like to thank Alison, my beautiful partner in life’s adventures; our children Nick and Daisy; plus our moggies Matilda and Rhubarb.

Publisher: Mike Sanders Associate Publisher: Billy Fields Acquisitions Editor: Susan Zingraf Development Editor: Kayla Dugger Design Supervisor: William Thomas Production Editor: Jan Lynn Layout: Ayanna Lacey Proofreader: Virginia V. Vought PRODUCTION, LONDON Digital Producer: Alex Valizadeh Senior Digital Producer: Miguel Cunha DIGITAL OPERATIONS, DELHI Head of Digital Operations: Manjari Hooda Producer: Rahul Kumar Assistant Editor: Etika Kapil DTP Designer: Manish Bhatt Operations Assistant: Tauhid Nasir

CONTENTS PART 1: TECHNIQUES The Principles of Modular Origami Assembling the Designs Creating Different Combinations

Choosing Your Paper How to Fold Paper Origami Symbols and Techniques Division Into Thirds The Iterative Method Folding Method 1 Folding Method 2 Folding Method 3

Producing 60-Degree Angles At the Center of the Paper At the Edge of the Paper

Using Templates Creating Third Creases Creating Quarter Creases

PART 2: DESIGNS

Diamonds

Versatile

Gem Flower

Eifel Star

Snap Hexahedron

Sonobe Unit

Dodecahedron

Jump Unit

Spinning Top

Woven Cube

Bow Tie Cube

Diamond Star

Truncated Cube

Sunken Octahedron

Segmented Unit

Trapezohedron

Bipyramid Blackberry

Truncated Star

Hinged Equilateral Unit

Bipyramid Dragon Fruit

Leaved Octahedral Skeleton

PART 1

TECHNIQUES

Before jumping in to the designs, I’d like to give you some practical and theoretical information that will enable you to get the most from this book. To begin, you learn about the ideas behind geometric origami. I also talk about what kind of paper is best for origami, as well as how to fold it. The section devoted to symbols is invaluable not only for the designs in this book, but also for any designs you decide to fold from other sources, such as the internet. With the folding techniques, you get a chance to go over the basic folds used in origami and to get used to the art of folding paper neatly and accurately. You also get information on dividing a square into thirds, creating a 60degree angle, and making a template, all of

which are important to know when working on the designs.

The Principles of Modular Origami

A modular design is one made from several sheets of paper, each folded identically with flaps and pockets that interlock to create a stable design. Flat forms of modular origami are usually made up of polygons (a plane shape with straight sides), while threedimensional versions are typically made up of polyhedra (a solid with flat faces).

The designs in this book are all modular; therefore, the following are principles you should keep in mind as you are assembling and combining. Assembling the Designs

Because modular designs aren’t made from a single sheet of paper like more naturalistic origami designs, the issue of assembly becomes of vital importance. With the majority of geometric designs, the structural integrity of the final design isn’t complete until the last unit is in place. Early stages of assembly can be frustrating—as you insert the sixth unit of 12, the first may slide out and before you know it, the whole assembly collapses. My advice for this is to use small amounts of masking tape or miniature clothespins to act as extra hands for you, holding units together until the end. It’s also true to say that your first attempt at a modular design may well look a bit scruffy by the time it is complete. The ideal scenario for all modular units is to handle them as little as possible, since the paper can quickly become “tired,” with the sharp creases becoming relaxed and not holding in place as effectively. When you come to make the

model for a second time, you’ll find your fingers know what to do and the model will be more accurate and neater and hence will hold together better. Creating Different Combinations One of the joys of modular designs is trying to find new ways to combine them. While you can approach this from a scientific perspective, it’s perhaps more fun simply to make lots and lots of units and just keep adding them to each other until they form a natural closure. Almost any triangularshaped unit can be combined in dozens of different ways, and almost any unit that creates a cube can also be combined in greater numbers to create stellated polyhedra. For example, if your modules are joined in a ring of three, see if you can form a ring of four or even five. Look for patterns in assembly—if you can see a pattern, you can repeat it, and there is every chance they will join together at some point. You can also try joining fewer units than the instructions suggest. For example, you can make a complete model using only three of the Segmented Units shown within this book. Another possibility is to turn units inside out and see how that works. Experiment, have fun, and don’t worry!

Choosing Your Paper

2D origami can be made using many different types of paper, but as soon as your subject becomes 3D, selecting the right material can make your life easier. In general, you want paper that is crisp (so it will retain creases), not too thick (since several sheets of paper may meet at a single point), and not too thin (or it might collapse under its own weight). After a while, you will be able to identify the qualities of a sheet of paper simply by folding a small corner over and examining the crease that is formed. You also should use paper that is square in order to get the best results. That doesn’t mean you can’t use standard printer paper or any other rectangular paper if it meets the previous requirements; it simply means you’ll need to cut the paper into squares.

Beyond type, think about what color of paper you’d like to use. While color combining is a subject too complex to be covered in any depth within this book, suggested color patterns are shown for each of the completed designs. Clearly, the more faces of a model, the more choices you have for combining the colors. If you’re interested in exploring color combining more, a quick search on the web for “polyhedral coloring” will give you lots of help. As well as colors, you can also choose from a rich variety of patterned paper to help your finished model stand out. Like with color, an internet search —in this case, for “modular origami”—will give you as much inspiration as you need! What’s Included with This Book This book includes folding papers to put you on your way to creating geometric origami! You can use them either to practice different folds or to make beautiful full designs. Want to incorporate other colors or patterns? No problem! You can combine the papers included with this kit with other papers you find elsewhere to create whatever looks are appealing to you.

How to Fold Paper

You might think it is the easiest thing in the world to fold a sheet of paper in half. However, to fold it perfectly in half requires some concentration and a little patience. The same is also true of all folds—any small error made at the start will seem to multiply as you get to the end of the sequence. There is no perfect approach that works the same with every fold; you will need to make adjustments. As long as the result is a neat crease, the method is good. However, the following walks you through one possible folding process to get you started: 1. Try to “see” the move before attempting it. Look at the next drawing and fold the paper in your mind before you begin to fold the actual paper. 2. Hold the paper firmly but not too tightly using both hands and move the paper to where it should lie. At this point, you can move the paper back and forth slightly, to be certain it is correctly positioned.

3. Use one hand to hold the paper firmly in position and make a crease using a finger of your other hand, as if you are “ironing” the paper flat. 4. Once flattened, you may wish to reinforce the crease, either with your fingers or a smooth plastic or wooden object (such as the end of a ruler). Some folders find tools work best, while others rely on their fingers alone —the choice is yours.

Whatever you decide to do, you need to be critical of your folding—if a step is not quite right, remind yourself to take a little more care the next time. Not only does accurate folding make the whole process easier, the final result will be more impressive as well. If you sense the paper is about to crumple or (even worse) tear, unfold immediately and start the fold again. With practice, you will find a way to work with the paper, rather than to force it into position. The paper sometimes seems to have a mind of its own, so you need to try to enter into a dialogue with it, rather than shout orders. Like any art form, origami can take a lifetime to master, so be patient and keep an open mind.

Origami Symbols and Techniques

One of the main reasons that origami has such an international appeal is because it uses a standard set of symbols to explain the folding sequence. While the accompanying text is useful, you can usually get by without it. This means you can follow origami diagrams even if you do not understand the language of the words. While you will find some small variations in the symbols used, once you understand them and the techniques behind them, you will only be reading the words from time to time. The following are the common symbols, as well as the folding techniques they represent.

Valley fold: Fold the paper the way you would normally do so.

Valley and unfold: Fold the paper, make a (firm) crease, and then unfold.

Mountain fold: Fold a flap or layer behind. This can be made as a valley fold by turning the paper over, as long as you remember to turn it back afterward.

Turn over: Lift the paper up and turn it over from side to side, like tossing a pancake.

Repeat: Whatever move you have made is repeated elsewhere on the paper. The arrow may also indicate which steps to repeat and/or how many times to repeat it. If you’re unsure of what to do, check the next diagram to see the result.

Rotate 180 degrees: Turn the paper in the direction of the arrow so the lower edge becomes the upper edge.

Rotate 90 degrees: Turn in the direction of the arrow so the left edge becomes the upper edge.

Fold to dotted line: A dotted line shows an X-ray of a hidden edge, which is used as guidance for a fold.

Pull out: Unfold or ease out flaps of layers to a previous position.

Push in (and X-ray): Using existing creases, apply pressure to push the paper inside itself (also known as an inside reverse fold); this is shown via a black

triangle. A dotted line may show an X-ray of where a hidden edge lies when the paper is pushed in.

Division Into Thirds

Many geometric designs will require you to divide the paper into thirds. There are several different ways of achieving this; the following are some methods you can try out. The Iterative Method Because it introduces minimal unwanted creases in the paper, many people prefer this method. It consists of guessing the distance, and then reducing the error.

1 Fold a corner up to an estimated third location, as

indicated by the dotted line. Before you make the crease, try to ensure the height of the folded flap is as close as possible to the height of the section that is left above it. When ready, make a very gentle and short pinch, and then unfold.

2 Fold the top corner to touch the first pinch and make a second, equally gentle pinch before unfolding.

3 Fold the lower corner to the latest pinch, making a fresh pinch that’s a little stronger than the first to distinguish the two, and then unfold.

4 Fold the top corner to the latest crease, making the

pinch more noticeable than the existing one, and then unfold.

5 Continue this sequence, until the latest pinches align with the previous ones.

Each time you do a pair of steps, the difference between your guess and the actual “one third” location (in other words, the error) will be halved. An experienced folder will often estimate so closely that their initial guess is almost perfect.

Folding Method 1

1 Start with a diagonal crease. Fold the right edge to the crease, and then unfold.

2 Fold the upper edge to the most recent crease, and then unfold.

3 Fold the top-left corner to the end of the most recent crease. This marks one third of the vertical edge.

Folding Method 2

1 Start with a diagonal and a halfway crease in place.

Make a new crease that joins the lower-left corner with the top of the vertical crease, and then unfold.

2 Make a vertical crease that passes through the

intersection of the two creases, and then unfold.

3 This makes one third of the lower edge. Folding Method 3

1 Start with a vertical halfway crease. Fold the lowerright corner to touch the upper end of the crease.

2 This is how your paper should look. Turn the paper over.

3 Fold the top-right corner to touch where the colored edge meets the white edge, and then unfold.

4 Unfold the flap from behind.

5 The horizontal line marks one third of the vertical edge.

Producing 60-Degree Angles

A great number of geometric models require you to create angles of 60 degrees. Luckily, there are a couple straightforward ways to do this, although they require accuracy and concentration. At the Center of the Paper

1 Start with both halfway creases and a left quarter

crease. Make a fold that passes through the center of the paper, such that the circled points meet.

2 This is how your paper should look. Unfold.

3 Here are two of the angles you have created, which includes a 60-degree angle at the center.

At the Edge of the Paper

1 Crease in half from side to side, and then unfold.

2 Fold, starting from the lower-left corner, so the lowerright corner lies somewhere on the vertical halfway crease. The idea is to try and “anchor” the lower-left corner, and then swing the paper around on that axis until it is in position.

3 Unfold.

4 Here are the key angles you have created, including a 60-degree angle positioned in the center of the right edge.

Dividing Into Thirds Creating a 60-degree angle at the edge of the paper can also be used to divide your paper into thirds. After you complete the previous steps, do the following:

1 Fold the lower-right corner to touch the circled intersection, crease, and unfold.

2 The lower edge is now divided perfectly into thirds.

Using Templates

The process of creating certain angles or dividing into thirds can produce creases that may not be needed on the actual model. This could be confusing to you as the folder —and may spoil the clean lines of the finished model. In order to avoid these problems, you can create a template for the angles or divisions. Here are two examples of templates you can make. Creating Third Creases

1 Using any of the methods outlined earlier, divide a

square (which is going to be your template) into thirds and fold one third over. Slide a clean sheet under the layer and make sure it is tight against the inside edge.

2 Fold the lower edge of the clean sheet to meet the raw (original) colored edge, crease, and unfold.

3 Remove the clean sheet.

4 Fold the upper edge to the existing crease, crease, and unfold.

5 You now have third creases with no other creases added.

Creating Quarter Creases

1 Fold the template sheet in half. Slide the clean sheet into it, making sure it is tight against the inside edge.

2 Fold the lower edge to the colored edge.

3 This is how your paper should look. Repeat three

times, rotating the inner sheet by 90 degrees each time. Remove the inner sheet.

4 You now have quarter creases with no other creases added.

Applying the Template Concept The Truncated Star design includes a sidebar showing a way in which the template concept can be applied to angles as well as divisions.

PART 2

DESIGNS

The theory behind origami you have absorbed from the previous part can now be put into practice with the following designs. The designs are presented in order of difficulty (although that in itself is subjective, depending on what skills you already have), so I recommend you follow the sequence— more or less! Remember, fold slowly and neatly, and don’t be afraid to throw a model away and start afresh if the paper becomes too tired.

Diamonds

Design by Robert Foord This diamond design is a classic example of geometric origami. It is essentially an octahedron with a flap on either side. While there seems to be no obvious reason why the two halves should interweave so effectively, they do—and that (apart from the clean lines and simplicity) is the joy of the design.

Construction

1 Start with the white side up. Fold in half upward, crease, and unfold.

2 Fold two opposite corners to the horizontal center crease, and then unfold.

3 Fold half of the upper and lower edges to the most recent creases.

4 Fold the lower-right corner to touch the inside colored corner (as circled), crease only in the lower half as indicated, and then unfold.

5 Repeat step 4 on the opposite side.

6 Fold in half upward.

7 Fold the lower-left corner back along an existing

crease, starting at the top-left colored corner. Repeat underneath.

8 Fold the white flap tightly over the colored edge, crease, and unfold. Repeat underneath.

9 Unfold the paper, except for the two small flaps.

10 Fold two corners in on existing creases.

11 Starting with the left side, fold so the circled points meet, creasing only in the lower half of the paper as indicated.

12 Repeat step 11 on the opposite side. Turn the paper over.

13 Convert the two short creases indicated from mountain to valley. Turn the paper over.

14 Form the paper into 3D using the creases shown.

15 The first unit is now complete. Make a second unit using a contrasting color.

Assembly

1 Arrange the units as shown and begin to interweave

them. The upper green flap lies outside the upper blue flap. On the hidden side, the opposite happens.

2 Continue to interlock, remembering to do the opposite on the hidden side.

Versatile

Design by Mick Guy This simple shape is formed by adding a triangle to a square. The fun is in making more than one and combining them into a pattern. It is a classic example of period tiling, where a shape can be used to cover a surface with no gaps.

1 Fold in half from edge to edge, crease, and unfold in both directions.

2 Fold an edge to the center, crease, and unfold. Repeat on the other three edges.

3 Fold in the lower-left corner to the nearest

intersection. Fold the right edge to the center.

4 Fold the upper edge to the center.

5 Fold the top-right flap over, crease, and unfold.

6 Ease out a corner from inside so the flap points upward.

7 Fold the flap down and to the right.

8 Fold the lower edge to the inner, colored edge.

9 This is how your paper should look. Turn the paper over.

10 Fold the lower-right corner up so the right edge meets the upper edge, crease, and unfold.

11 Fold the upper-right corner down so the right edge lies along the lower edge.

12 Fold the upper half behind.

13 Fold the inner colored corner upward as you allow the right corner to move inward.

14 Fold the white triangle in half upward while pressing the lower-right corner to create a colored triangle.

15 Tuck two flaps into their respective pockets.

Combining Versatiles One way you can challenge yourself is to find a different way to use versatiles to create the same shape. Is the paper used efficiently—in other words, can you make a larger version from the same-size square? Does your version “lock” the edges together? The following are three possible combinations with the same shape—a square—at their center. What others can you create?

Gem Flower

Design by Nick Robinson The flower and stem of this design fit together as if they were meant to. While not a specific geometrical form, this is an example of how origami can represent nature in a geometric way.

Flower Construction

1 With the colored side up, crease in half corner to

corner and then unfold in both directions. Turn the paper over.

2 Fold in half side to side, crease, and unfold in both directions.

3 Fold the lower edge to lie on a diagonal, creasing only from the corner to the diagonal.

4 Repeat with a mirror-image fold.

5 This is how your paper should look. Repeat steps 3 and 4 on the other three edges.

6 Convert the outer sections of the diagonals to valley folds. Turn the paper over.

7 Convert the outer sections of the halfway creases to valley folds.

8 Start to push in these creases; don’t add any new

ones! The four corners should move toward each other underneath.

9 This is the move in progress.

10 The flower unit is complete. Determining What Stem Pieces Are in Color Starting with the paper white side up gives the stem a colored outer surface. However, you also have the option of starting with the colored side up if you want the inner leaf faces to be colored.

Stem Construction

1 Start with the white side up. Crease in half corner to corner and then unfold in both directions. Turn the paper over.

2 Fold in half upward.

3 Fold each half of the lower edge to the diagonal creases, and then unfold.

4 Push the corners inside on the previous creases.

5 Unfold downward.

6 Fold in half from left to right.

7 Repeat steps 3 through 5 three times.

8 This is how your paper should look. Turn the paper over.

9 Start to gently press the creases in, raising the sides toward you.

10 The stem unit is complete.

Assembly

Arrange the flower above the stem, and then slide the two together. The inner flaps of the stem slide into the narrow pockets of the flower. Tighten the two halves.

Eifel Star

Design by Hans-Werner Guth The Eifel is a low mountain range in western Germany. The elegant modular design that gets its name from it is based around a simple 60-degree crease pattern. Every crease is perfectly located and nothing is forced into place. And with 12 units, you can try many different color combinations to really make it pop.

Construction

1 Start with the colored side up. Fold in half from side to side, but only crease in the upper quarter as indicated.

2 Starting the crease at the lower-right corner, fold so the lower-left corner lies somewhere on the vertical halfway crease.

3 Fold the inner long white edge to the outer edge.

Using a Template for New Units Once you’ve done step 2 on your first unit, you could use a sheet as a template to bypass step 1 when creating the rest of your units.

4 Fold the right edge to the lower-left edge.

5 Fold the raw (original) white edge to the upper-right edge.

6 This is how your paper should look. Turn the paper over.

7 Fold the paper in half downward.

8 Fold over an inner colored edge and crease firmly.

9 Unfold back to a colored square.

10 Use the existing creases to mountain fold the outer flaps behind.

11 Fold the sharp corner to lie on the intersection of the creases (as circled).

12 Reinforce the mountain creases, and then make the indicated fold over and fold behind simultaneously.

13 The first unit is the complete. Make another 11 units following the previous steps, using three sheets each of four colors (or any other combination you wish).

Assembly

1 Arrange two units and slide the blue one into the center of the yellow one.

2 This is an alternate view of the assembly. The yellow unit slides under the central flap of the blue unit.

3 Reform the creases made in step 12.

4 Carefully tuck the yellow flap underneath the blue pocket below it. Repeat on the underside.

5 The two units are now fully interlocked.

6 Arrange a third unit and slide the yellow flap inside, as you did previously.

7 Tuck the blue flaps under the yellow layers. Continue with the remaining units.

Snap Hexahedron Design by Larry Hart

With the majority of geometric designs, you get a feel for the final design long before it is complete. This six-sided design (which is also known as a triangular bipyramid) keeps its secret until the final move, where it magically transforms from 2D to 3D! This “climactic” model works best when made with crisp paper.

1 Start with the white side up. Fold in half, crease, and unfold in both directions. Turn the paper over.

2 Fold an edge to the center, crease, and unfold. Repeat on the other three edges. Turn the paper over.

3 Fold so the circled points meet, creasing only within a small square. Unfold and repeat on the other three corners. Turn the paper over.

4 Fold along the center, changing the section indicated to a valley crease.

5 Fold the upper half behind and press in the right and left edges to form a square in the center. The paper should become 3D.

6 Pressing the sides of the central square together,

swing the nearest flap to the right and the matching flap underneath in the opposite direction.

7 Fold the lower-right corner to the center, crease, and unfold.

8 Lift a single layer upward, flattening the lower-right corner into a triangular flap.

9 Fold the lower-left corner and all layers at the top-left corner to the center.

10 Lift up the center of the triangular pocket on the right and fold the triangular flap from the left underneath it.

11 This is how your paper should look. Turn the paper over.

12 Repeat steps 7 through 11 on this side.

13 This is how your paper should look. Rotate the paper clockwise by 90 degrees.

14 Press at the top and bottom; the paper will “snap” into 3D.

Sonobe Unit

Design by Mitsunobu Sonobe This versatile unit (or module) was created in the late 1960s and has proved highly popular, since it is easy to fold, easy to assemble, and allows for many different combinations. As you will see, it is also possible to adjust the folding to create many different patterns. As you try the different sonobe combinations, remember to fold neatly at all times, so your final result is crisp and accurate.

1 Start with the white side up. Fold in half from top to bottom, and then unfold.

2 Crease and unfold upper and lower quarter creases.

3 Fold the top-left and bottom-right corners to the nearest crease.

4 Fold the lower edge to the center.

5 Fold the left vertical edge to lie on the upper quarter crease.

6 Fold the upper edge to the original center crease.

7 Fold the right vertical edge to the lower edge, crease, and unfold.

8 Lift the colored edge up slightly.

9 Fold the top-right corner into the pocket.

10 This is how your paper should look. Turn the paper over.

11 Fold the lower-left corner to the top-left corner, crease, and unfold.

12 Fold the top-right corner to the lower-right corner, crease, and unfold. Turn the paper over.

13 The unit is complete. This shows the flaps and the

pockets into which the flaps will fit (on adjacent units).

Sonobe Hexahedron Also known as a dipyramid, this is the simplest three-dimensional form that can be made with the unit and requires three sonobe units. Choose different colors for an enhanced visual effect and to help you see how the design fits together.

1 Arrange a unit as indicated from the underside. Fold in half from side to side, crease, and unfold. Repeat with the other units.

2 Turn the units over and arrange two as indicated. Slide the blue flap into the green pocket.

3 Fold the green unit behind using a mountain fold.

4 Tuck the lower-green flap into the blue pocket. Flatten the paper firmly.

5 Press gently as indicated to open the paper into 3D.

6 Take the third unit and feed the yellow flap into the blue pocket.

7 Tuck the green and blue flaps into the yellow pockets.

8 Fold the loose yellow flap behind and tuck it into a

green pocket. Fold the flaps over to 90 degrees. Arrange two units as indicated and feed the yellow flap into the green pocket.

Sonobe Cube A sonobe cube requires six sonobe units— one for each face of the cube. It is easiest to use three colors, with two units of each color. Note that you do not need the extra crease the sonobe hexahedron begins with.

1 Fold the flaps over to 90 degrees. Arrange two units as indicated and feed the yellow flap into the green pocket.

2 Arrange a third unit above the other two. Feed the

blue flap into the yellow pocket and then the green flap into the blue pocket.

3 Roll the nearest side to face downward.

4 Add the fourth unit in the color that matches the

opposite side. Slide the green flap into the blue pocket, and then the yellow flap into the green pocket.

5 Roll the right side to face downward.

6 Add the fifth unit in the color that matches the

opposite side. Tuck the yellow flaps into the green pockets on two opposite sides. Once that’s done, tuck the blue flap into the yellow pocket.

7 Roll the nearest side to face downward.

8 Add the sixth unit. The blue flap goes into the yellow pocket, and then the green flap goes into the blue pocket.

9 Tuck the green flap into the blue pocket, and then

complete the cube by tucking the blue flap into the yellow pocket. Move the cube around in your hands, tightening everything up.

Dodecahedron

Design by Boaz Shuval While it is relatively easy to produce 45- and 60-degree angles in origami, the creation of a 108-degree angle (the internal angle of a regular pentagon) is more challenging! The resulting face is shown in grey on the illustrations—try not to allow any creases to extend into this area.

Construction

1 Start with the colored side up. Crease and unfold a diagonal. Turn the paper over.

2 Crease and unfold the lower half of the other diagonal.

3 Fold the right edge to the diagonal and crease only at the lower end before unfolding.

4 Make a vertical crease that passes through the lower

end of the most recent crease. Make this crease only in the area above the highlighted pentagon.

5 Fold so the crease starts at the lower-left corner and the lower-right corner lies somewhere on the most recent crease. Take your time to be accurate.

6 This is how your paper should look. Turn the paper over.

7 Fold through the original diagonal so the crease is transferred to the lower layer. Turn the paper over.

8 Unfold the colored flap, putting the mountain crease in place.

9 Fold the lower-left corner to the upper-right corner but don’t let the crease extend to halfway; keep the upper half curved.

10 Fold the white flap over the colored edge, and then unfold the large flap.

11 Convert the diagonal into a valley by folding upward.

12 Fold the white flap over the colored edge.

13 Repeat on the other side.

14 Unfold the two flaps, open out the inner layer, and then refold the upper flaps.

15 Fold again on the diagonal, interlocking the two central flaps.

16 Fold the lower-left corner to touch the end of the crease, crease, and unfold.

17 Fold the right corner to touch the upper end of the most recent crease and then unfold. Turn the paper over.

18 The first unit is complete. Make 11 more identical units following the previous steps, using two sheets each of six different colors or three sheets each of four different colors.

The Makeup of the Dodecahedron Unit The unit should have two flaps, two pockets, and (hopefully) no creases on the face. This means the final model will have some edges that do not interlock, but this is not a problem.

Assembly

1 Interlock four units. Two adjacent edges (green and blue) do not interlock.

2 Add the next unit, interlocking two flaps and pockets.

3 The next unit has one interlocking flap/pocket, with the green/red edges not interlocking. Continue interlocking the remaining six units, producing an identical pattern on the hidden side.

Jump Unit

Design by Hans-Werner Guth This intriguing design is a rarity in the world of origami because it uses curved creases. The best way to add these is to find a circular object and a ballpoint pen that has run out of ink, so you can literally “draw in” (or score) the creases. The design works best with thick paper or thin card stock. Experiment to see how your chosen material allows you to add the curved creases.

Construction

1 Start with the white side up. Fold in half upward, make small pinches at the edges, and unfold.

2 Repeat in the opposite direction.

3 Using an old CD or any other circular object, score a curved crease with your pen.

4 Repeat at the next corner. Repeat for the other two corners.

5 This is how your paper should look. Turn the paper over.

6 Carefully emphasize the creases as mountains.

7 The first half of the unit is complete.

8 With another square of the same size white side up, crease and unfold both diagonals.

9 Using the same circular object and your pen, score a curved crease. Repeat on the other three corners.

10 This is how your paper should look. Turn the paper over.

11 Pinch the scored lines into mountain creases.

12 The second half of the unit is complete.

13 Arrange the two halves as indicated and interlock the two so the corners of the upper unit lie on the diagonals of the lower unit.

14 Holding the two sheets tightly together, wrap the lower green corners over the edges beneath. Make these creases firm.

15 Press the corners together and re-emphasize the

mountain creases, encouraging the lower edges to curve inward, so the corners angle downward. Repeat on the other three corners.

16 The first full unit is complete. Make five more, using either six different colors or two each of three colors.

Alternate Method for Scoring When you are familiar with the scoring method outlined in the construction steps, you can try this alternate approach. Start with both sheets just before the scoring is added, and then precisely overlap them as indicated in the following image, so the diagonals of the lower sheet align with the halfway pinches of the upper. Now score as shown. You want to apply just enough pressure to pass the crease through to the lower layer, but not so firmly that the upper layer is scored too strongly. You should then be able to complete steps 14 and 15.

Assembly

1 Arrange two units as indicated. Slide a light blue flap

above the adjacent flap but underneath the upper layer—it doesn’t matter which goes under which. Align the mountain creases on both sheets.

2 Interlock the third unit with both existing units. Again, slide either flap into the adjacent pocket.

3 Here are three units combined. Add the others in the same way. Having a helper lend you a third hand will greatly assist this process. You may find flaps tend to come out until the final unit is in place, but be patient and gentle—you can tighten them all at the end.

Spinning Top Design by Various

Some origami designs are the result of a series of adaptations and extensions, so they do not have a single creator. This brilliant design was developed by several folders and is made up of three parts, with the smallest part being quite thick. However, since it is a simple fold, it’s not a big problem. The end result is solid and spins very well!

Part 1 Construction

1 Start with the white side up. Crease in half from side

to side, and then unfold. Repeat in the other direction.

2 Fold in half diagonally, crease, and unfold in both directions.

3 Fold all four corners to the center (a move known as a blintz).

4 Again, fold all four corners to the center.

5 Again(!), fold all four corners to the center.

6 Flatten the paper into a waterbomb base.

7 Part 1 of the spinning top is complete. Part 2 Construction

1 Complete steps 1 through 3 of Part 1 Construction. Turn the paper over.

2 Fold all four corners to the center.

3 This is how your paper should look. Turn the paper over.

4 Fold all four corners to the center.

5 This is how your paper should look. Turn the paper over.

6 Fold the four inner corners out to the outer corners.

7 Part 2 of the spinning top is complete. What Is a Waterbomb Base? A waterbomb base is a base formed by bringing the midpoints of the four edges of a square together.

Part 3 Construction

1 Complete steps 1 and 2 of Part 1 Construction. Fold

each corner to the center, crease, and unfold. Turn the paper over.

2 Fold each edge to the halfway crease, and then unfold.

3 Collapse the paper inward using only the existing creases.

4 Fold the inner raw (original) edges of the lower-left square to the diagonal creases, and then unfold. Repeat on the other three squares.

5 Inside reverse the corners of the lower-left square. Repeat on the other three squares.

6 Fold a flap out. Repeat on the other three flaps.

7 Fold a flap up as far as it will go. Repeat on the other three flaps.

8 Fold the narrow flap in. Repeat on the other three narrow flaps.

9 Part 3 of the spinning top is complete.

Assembly

1 Keeping part 1 slightly 3D, tuck the flaps into the pockets of part 2.

2 Carefully slide the part 2 corners under the part 3 flaps, folding slowly and trying not to add any new creases.

3 This is how your paper should look. Turn the paper over.

4 Gently emphasize the creases indicated, so the top has a central point to spin on. Turn the paper over.

Woven Cube

Design by Nick Robinson Most modular origami cubes are created using one square for each face of the cube, requiring six units. This design is no exception and only requires a few simple creases. The end result holds together well and, as you will see, has numerous possibilities for many different combinations.

Construction

1 Start with the colored side up. Fold in half from

bottom to top, making small pinch marks, and then unfold.

2 Fold in half from side to side, crease, and unfold.

3 Fold the left and right edges to the center crease. Turn the paper over.

4 Fold the left edge to the center crease, allowing a flap to flip out from underneath.

5 This is how your paper should look. Repeat on the right side.

6 Fold the upper and lower edges to the center.

7 This is how your paper should look. Open the flaps out halfway; the first unit is complete. Make another five units following the previous steps, using two sheets each of three different colors.

Assembly

1 Take two units and slide the flap of the green unit

behind a layer of the yellow unit. Unfold the flaps of the yellow unit if it helps.

2 Take a third unit and slide the purple flap into the

green pocket while sliding the yellow into the purple at the same time. You can assemble loosely, and then tighten up around the common corner.

3 Repeat the same process on the underside, so the opposite-side faces are made from the same color.

Bow Tie Cube

Design by Vignesh Cumareshan This design was inspired by a class where the Woven Cube was taught and clearly shows how origami builds and expands upon existing work, allowing you to discover something new. In this case, a few extra creases in the center allow the cube to be extended into a more interesting, richer form.

1 Start with the colored side up. Fold in half from to top to bottom, crease, and unfold.

2 Fold in half from side to side, crease, and unfold.

3 Fold the left and right edges to the center crease. Turn the paper over.

4 Fold the left edge to the center crease, allowing a flap to flip out from underneath.

5 This is how your paper should look. Repeat on the right side.

6 Fold the upper and lower edges to the center.

7 This is how your paper should look. Open the flaps out halfway.

8 Make diagonal creases at each corner. Open the

model back to the square and turn the paper over.

9 Add the four creases indicated. Turn the paper over.

10 Fold so the circled points meet and crease only in the central area.

11 Repeat on the other side.

12 Fold the right edge over so the circled points meet.

13 Fold so the creases on the lower layer are transferred to the upper layer.

14 Unfold the flap.

15 Repeat steps 12 through 14 on the left side.

16 Alter the creases indicated so they are valleys on the white side.

17 Reform the existing pleats.

18 The first unit is complete. Make another five units

following the previous steps, using two sheets each of

three different colors. Combine the six units according to the Woven Cube assembly steps.

19 Carefully press the sides of the central bar together, forming a valley crease at the center on either side.

20 Repeat on the other two sides, and then repeat in the same way on the underside.

Diamond Star

Design by Francesco Guarnieri Stars are beautiful shapes, and origami offers you hundreds of different 2D designs to choose from. However, this version gives you a star that’s interestingly 3D. The geometry of this design allows the five units to fit together perfectly.

Construction

1 Start with the white side up. Fold in half from right to left diagonally.

2 Fold the lower-left edge of the first layer to the vertical edge. Repeat underneath.

3 Unfold from the center to the white side.

4 Fold the top corner to the left corner (as circled),

crease, and unfold. Repeat in the opposite direction.

5 Fold the lower corner to touch the right corner, crease only in the left half as indicated, and then unfold. Repeat in the opposite direction. Turn the paper over.

6 Make a similar fold to the previous, but this time

crease only in the right half. Repeat in the opposite direction.

7 Open out the white flaps.

8 Fold the top corner to the circled intersection, crease, and unfold.

9 Fold so the circled points meet and crease only where indicated. Turn the paper upside-down from top to bottom.

10 Fold in half to change the indicated creases to valleys.

11 Reinforce the indicated creases and begin to form the paper into 3D.

12 Fold the lower-left edge inward using the creases indicated. The paper stays 3D from here onward.

13 Rotate a colored square area of paper to the left, to lie within the model.

14 Carefully press the white edge inside the pocket.

15 Lift the lower-right corner upward using the creases indicated.

16 Fold the upper-right corner to touch the lower

corner. At the same time, gently press the white area inward to match the other side.

17 This is how your paper should look. Turn the paper over.

18 The first unit is complete. Make four more units following the previous steps.

Assembly

1 Slide the flap of the left unit into the outer of the two pockets of the right unit.

2 Add a third unit in the same way. Continue with the remaining units.

Practice Makes Perfect While the sequence is not especially difficult to fold, you may need to make a few test versions before your result is cleanly folded and impressive. Origami requires practice to perfect, and this design is a good example of how much practice may be needed!

Truncated Cube Design by Various

The unit for this design is based on a traditional design called a Yakko San and forms a cube with the corners “sliced off,” or truncated. In theory, the flaps and pockets are the same size, but unless the paper is very thick, there is enough flexibility for one to slide into the other.

Construction

1 With the white side up, fold in half side to side, crease, and unfold in both directions.

2 Fold all four corners to the center.

3 This is how your paper should look. Turn the paper over.

An Added Challenge Try making the creases at step 1 only on the outer quarters, so the central square of the final unit is “clean.” The references will still be there if you look for them.

4 Fold all four corners to the center, crease, and unfold. Turn the paper over.

5 Fold the lower corner to the opposite quarter crease. Crease the central half of the inner square as indicated, and then unfold. Repeat on the other three corners.

6 Fold the corners behind on the existing mountain creases.

7 Fold all four corners to the center.

8 This is how your paper should look. Turn the paper over.

9 Open the lower raw (original) edges, folding the inner corner down and squashing the flaps flat.

10 This is how your paper should look. Repeat on the other three edges.

11 This is how your paper should look. Turn the paper over.

12 Fold the lower edge to the center, crease, and

unfold. Repeat on the other three edges. The first unit is complete. Make five more units, using two each of three colors.

Assembly

1 Arrange two units as indicated. Slide the green flap into the matching pocket of the blue unit.

2 Add a third unit by inserting the green flap into the blue pocket.

3 Add the fourth and fifth units by sliding two blue flaps on each side into the yellow pockets.

4 Add the blue flap of the final unit into the yellow pocket on the left.

5 Lift up the left and right units, forming them into four

sides of the cube. The blue flap from the left tucks into a yellow pocket from the right.

6 Tuck the yellow flaps into the green pockets and the

green flap into the blue pocket. Repeat on the opposite side.

Sunken Octahedron Design by Nick Robinson

There are many different ways to create this shape through origami, ranging from a complex single-sheet method to a simpler six-sheet method. This solution uses two sheets, ideally of contrasting colors, in which two triangular flaps from each side can be seen on the other half of the finished model.

Construction

1 Start with the white side up. Fold in half from one

corner to the opposite corner and then unfold. Do this in both directions.

2 Fold a corner to the center, crease, and unfold.

Repeat on the other three corners. Turn the paper over.

3 Fold in half from one side to the opposite side, crease, and unfold in both directions.

4 Fold an edge to the center crease, and then unfold. Repeat on the other three edges.

5 Fold so the circled corner meets the circled

intersection, but crease only where indicated. Unfold and repeat on the other three corners. Turn the paper over.

6 Carefully change these new creases so they become valley folds. Turn the paper over.

7 Press the sides together so the model becomes 3D.

8 Fold on the short, outer valley creases, slightly

rotating the paper in a counterclockwise direction. No new creases are needed.

9 This is the fold in progress. Do not add any new creases, and try not to force the paper!

10 When the twist is complete, sharpen up the creases, and then turn the paper over.

11 Fold the center white triangle in half to the right. The upper corner should naturally fold downward.

12 Fold the green corner to the right, behind a layer, as you squash the top-left corner.

13 Repeat steps 11 and 12 on the lower-right corner.

14 Fold the outer colored triangles in half, crease, and unfold.

15 Reverse the colored corners inside.

16 The first unit is complete. Make a second unit following the previous steps.

Assembly

1 Arrange the two units with one facing up and the

other facing down. Bring them together so the central square sections meet.

2 Fold the lower flap over, tucking the end of it into a pocket.

3 Repeat on the matching flap behind.

4 Fold the sides underneath, tucking the flaps into the pockets in the same way you did previously.

What Makes This a “Sunken” Octahedron? An octahedron is composed of eight equilateral triangles, four of which meet at each vertex.

Here is one face of an octahedron. The center is where the dotted lines meet.

A sunken octahedron is made by taking the center of each face and moving it inside until it touches the center of the polyhedron.

Segmented Unit Design by Dave Brill

This clever design pleats the paper into a 3×1 rectangle, which produces pockets down either side and flaps at either end. Because the geometry of a 3×1 isn't perfect for equilateral triangles, the end flap is slightly longer. Thankfully, this flap gets hidden. The assembly is quite loose to begin with but tightens up as you add more units.

Construction

1 Start with the white side up. Divide into thirds (see

Part 1 if you need a refresher on this). Pleat the paper on the creases.

2 Fold the upper layer in half, crease, and unfold.

3 Starting at the upper-right corner, fold the upper-left corner to lie on the vertical crease.

4 Fold the colored triangle in half, crease, and unfold.

5 Unfold the colored triangle.

6 Starting at the circled point, fold so the lower-left edge lies along a crease.

7 Fold so the upper colored edge lies on the lower-left edge.

8 Fold upward along a hidden edge.

9 This is how your paper should look. Unfold the last step.

10 Fold the short, white, folded edge to lie on the crease.

11 Fold the colored flap in half, crease, and unfold.

12 Unfold so the colored side is revealed.

13 The first unit is complete. Make five more units in any color scheme you’d like.

Assembly

1 Combine two units by sliding the green unit into the yellow unit so the creases coincide.

2 This is how your paper should look. Emphasize the creases indicated so the paper becomes 3D.

3 Rotate the units and add a third unit in the same way, joining the existing two units.

4 The three units should form a triangular hole in the

center. Add a fourth unit, so the yellow tucks into the orange and the green tucks into the yellow.

5 Rotate the model counterclockwise.

6 Add a fifth unit by sliding the green unit into the

yellow and the orange into green. Add the final unit at the back of the model in the same way. Turn the paper over.

7 The three “loose” ends can now be joined to one another.

Other Combinations The following are some other designs that can be created with the segmented unit. Feel free to try these or any other combinations to test yourself!

This design uses 12 units.

This design is made up of 30 units.

Trapezohedron

Design by Nick Robinson A trapezohedron is a crystalline form whose faces are trapeziums. This design is produced using eight flat modules. Each half of the design is a mirror of the other that can be locked together. The joining mechanism relies on tension in the paper to hold the flaps in the pockets.

Construction

1 With the white side up, fold in half vertically, crease, and unfold.

2 Fold the left and right edges to the vertical center, crease, and unfold.

3 Make a crease starting from the center of the lower

edge so the lower-right corner lies somewhere on the

left three-quarter crease.

4 Fold so the upper colored corner touches the lower colored corner.

5 Fold the lower-right colored corner to the top-right corner.

6 Unfold the paper fully.

7 Make a parallel pleat using the two creases indicated.

8 This is how your paper should look. Turn the paper over.

9 Fold the two small triangles tightly over the adjacent edges, and then unfold.

10 Fold the same flaps over, tucking them under a layer. Turn the paper over.

11 Unfold a double layer.

12 Fold the right quarter inward.

13 Fold the upper-right edge downward.

14 Fold through a crease on the lower layer, crease, and unfold.

15 Fold the top-left corner to the lower inside colored corner.

16 Fold the triangle in half, crease, and unfold.

17 Lift up the triangular flap.

18 Fold the left vertical edge to the right one-quarter vertical edge/crease.

19 Fold through a crease on the lower layer, crease, and unfold.

20 Swing a flap to the left.

21 Inside reverse the lower-left corner on the existing creases.

22 Fold the lower edge to the nearest colored edge,

crease, and unfold. Fold the white triangle to the left, crease, and unfold.

23 The first unit is complete. Make three more in this fashion using whatever colors you’d like. Make another four units that are mirror images of the first four, substituting “left” for “right” and vice versa in the steps!

Assembly

1 Arrange two same-style units as indicated and slide the blue unit into the green unit.

2 Lock the units by folding over the white triangle and tucking it under a layer.

3 Add a third (yellow) unit.

4 Lock in the same way, with the white triangle of the second unit folding over and tucking under a layer.

5 Add and lock a fourth (purple) unit, and then insert the first (green) unit into the fourth and lock.

6 The first set of four is complete. Make another set using the four mirror-image units.

7 Arrange both halves as indicated, and then carefully

interlock so a flap from each half tucks into a pocket on the other half. Begin to interlock them and then slowly rotate each half in opposite directions, feeding the flaps fully into the pockets. Go carefully!

Bipyramid Blackberry Design by Denver Lawson

The basis for this design is known as the “salt cellar” or “fortune teller.” The designer wanted to join several of these pieces together and, in a flash of inspiration, folded each one from two sheets, meaning each corner could tuck into another corner.

Construction

1 Start with two sheets, both white side up. Arrange the sheets so they line up exactly. From here onward, treat both sheets as if they are a single sheet.

2 Fold in half from side to side, crease, and unfold in both directions.

Salt Cellars and Fortune Tellers While you may not recognize the terminology, the unit will probably be very familiar to you—perhaps you’ve even created one yourself! With salt cellars, the four points sit face down and the pockets are filled with small pieces of food. With fortune tellers, the four points are face up, with the inside faces and flaps labelled with different information to reveal a person’s “fortune.”

3 Fold in half from corner to corner, crease, and unfold in both directions.

4 Fold all four corners to the center.

5 This is how your paper should look. Turn the paper over.

6 Fold all four corners to the center.

7 Fold in half from side to side, crease, and unfold in both directions. Turn the paper over.

8 Fold in half from corner to corner, crease, and unfold in both directions.

9 Fold the corners so they meet underneath. The paper should form into 3D.

10 Ease out both layers of each corner using a mountain crease.

11 The first unit is complete. Make six in total, either two each of three colors or one each of six colors, following the previous steps.

Assembly

1 Arrange two units as shown and slide the blue flap in between the two layers of the green unit. Do not slide in fully.

2 This is how the underlying structure should look so far.

3 Add a third unit, represented in yellow. A green flap slides between yellow layers and a yellow flap slides between blue layers.

4 This is how the model should look after tightening up the three units.

5 This is how the underlying structure should look. Add three more units in the same way, so the matching colors are opposite one another.

Truncated Star Design by Francis Ow

This design utilizes a 120-degree unit created by Ow, and adds the truncating element you learned about in Truncated Cube. Assembly is easy enough to begin with, but the more units you add, the harder it is to tuck the flaps into the pockets. Simply use your fingers, a thin stick, or even glue to lock the units.

Construction

1 With the white side up, crease in half from side to side, and then unfold. Turn the paper over.

2 Fold the sides to the center crease.

3 Fold the inner corners to the outer corners, making a short but firm pinch, and then unfold.

4 Unfold the flaps.

5 This is how your paper should look. Turn the paper over.

6 Fold, starting from the lower-right pinch, so the lower-

left pinch lies somewhere on the vertical center crease. Take your time and be accurate!

7 Fold the colored corner down using existing creases;

at the same time, swing a flap from left to right. Flatten the paper.

8 Crease firmly, and then unfold. Repeat steps 6 through 8 on the opposite corner.

9 This is how the crease pattern should look so far.

10 Fold the bottom-left and upper-right corners over to lie along the nearby short creases.

11 Fold the lower colored edge up on an existing crease.

12 Repeat step 7. Template Shortcut Because you will need to fold many units, there is a potential shortcut to this pattern using a template. Fold to step 7, and then position another sheet from step 5 underneath it, lined up exactly. Fold the lower sheet tightly over the existing crease, and then repeat on the other side.

13 Fold on an existing crease, reversing the top-left corner.

14 Repeat step 7 at the top.

15 Fold the opposite corners to the center, crease, and unfold.

16 Fold the vertical raw edges to lie on the short

creases, crease, and unfold. Turn the paper over.

17 Refold the central crease through all layers, and then unfold. Turn the paper over.

18 The first unit is complete. Make 11 more units

following the previous steps in any color you’d like.

Assembly

1 Arrange two units as indicated, and then slide the blue flap behind a layer of the orange unit.

2 Fold the blue flap back over, tucking it behind the orange layer. The two units are now assembled.

3 Add a third unit.

4 Lock it as you did with the previous unit.

5 Flatten the three units together, pressing them firmly in place. Allow the units to unfold naturally.

6 Add and lock a fourth unit.

7 This is how your paper should look. Turn the paper over.

8 Refold the recent creases, so the sides of the model face downward. Press the inner edges together to reinforce the shape. Make an identical form to this using four more units that will be used later.

9 Add a fifth unit, joining the lower corners of the orange and blue units.

10 This is how your paper should look. Add two more units on either side, and then a fifth at the back, forming a horizontal ring around the upper section.

11 Turn the second half (made at step 8) upside-down and interlock it with the model using the previous locking methods.

Hinged Equilateral Unit Design by Nick Robinson

The equilateral triangle is perhaps the most flexible shape when it comes to modular designs. This unit combines two adjacent triangles with a central hinge that can be folded either way. Using this unit, you can make a number of combinations, three of which I will show you here.

Unit Construction

1 With the colored side up, crease in half side to side, and then unfold. Turn the paper over.

2 Crease in half from top to bottom, and then unfold.

3 Fold the top and bottom edges to the horizontal center, crease, and unfold.

4 Make a fold starting at the center of the left edge, so the lower-left corner lies somewhere on the upperquarter crease. Do not extend the crease above the vertical center crease.

5 Crease only where indicated, and then unfold. Rotate the paper counterclockwise.

6 Repeat step 5 with the left corner, and then on each corner of the upper half.

7 Fold the sides inward.

8 Fold along a hidden crease as indicated to add it to the upper layer. Repeat on the other three corners.

9 Unfold the flaps and turn the paper over.

10 Fold in half upward.

11 Fold a short layer over, with the crease passing through two intersections; note how the angled creases line up.

12 Fold the lower flap down underneath, leaving the pleat in place.

13 Carefully fold along the crease line indicated and unfold. Repeat in the opposite direction.

14 Fold the outer sections down, and then unfold.

15 Fold the left edge inward, swiveling a flap within on the short valley crease. The mountain crease should be formed as you flatten that area.

16 Repeat step 15 on the right edge.

17 Reverse two opposite corners inside, so the white sections are exposed.

18 At the lower end, fold the white flap over the

colored edge, crease, and unfold. Fold the lower edge to the white edge, crease, and unfold. Repeat on the upper half. Turn the paper over.

19 The unit is complete. Tetrahedron For this design, you need to add a couple more creases to one of the two units used. Otherwise, you won’t be able to fit the units together properly for this design.

1 Tuck the white flaps of a unit underneath.

2 Fold the upper and lower edges firmly to the nearest crease.

3 Make a second unit that’s identical to the first unit except at step 15 of unit construction, reverse the other corners inside to create a mirror-image unit.

4 Partially form the creases indicated on the first unit. Repeat on the second unit.

5 Arrange the units as indicated, so one is rotated 90

degrees to the other. Put the flaps from each unit into the pockets on the other.

Triangular Dipyramid You need three of the basic units to create this design.

1 Arrange two units as indicated, and then slide the yellow flap into the green pocket.

2 Ensure the two units are perfectly positioned before

folding the white flap of the green unit over and under a layer to trap it.

3 Slide in a third (blue) unit.

4 Lock the second and third units as you did the first and second.

5 Fold along the valley creases to make the model 3D. Carefully slide the green flap inside the blue pocket.

6 Using care, lock the rest of the units together in the

same way. You may need a thin tool to get deep into the model, but lifting up the outer layer before folding should help.

7 Join the outer flaps in the same way you joined the

previous ones. You may be able to lock one of these pairs, but it shouldn’t be necessary if you have made sharp, accurate creases.

Stellated Octahedron Picture an octahedron where each (triangular) face has a three-sided pyramid attached, and you have a “stellated” octahedron. While 8 faces multiplied by 3 triangles equals 24 triangles, each sheet provides two. Therefore, you only need 12 units in total.

1 Fold one unit to the end of step 4 of the Triangular

Dipyramid and view it from the outside. The threesided pyramid will form one of the outer stellations.

2 Make another unit in a mirror image of the color combination for the first unit.

3 Arrange the two units as indicated and interlock the

two flaps that form a horizontal line at the top. Ensure the units are tight together before locking; care taken here will be repaid later on.

4 Add four more units in the same way until six units are joined as shown. Make another combination following the previous steps.

5 Rotate the lower half by 90 degrees, and then

carefully and patiently interlock the two halves, locking inside where possible. A thin stick can help you “flip the flaps” inside almost to the end.

Bipyramid Dragon Fruit Design by Nick Robinson

This design is based on the techniques developed by Denver Lawson for his Bipyramid Blackberry. Like that design, it uses two sheets to provide locking pockets. The division of the side into thirds produces smaller bipyramids and an underlying geometric structure of a stellated octahedron.

Dividing Into Thirds If you are not confident about how to divide each square into thirds, you can try the method of using a folding template outlined in the steps. If you know other methods, add the thirds and then jump to step 4.

Construction

1 Arrange two squares with the white sides together.

From here onward, treat both sheets as a single sheet (and try to keep them lined up neatly).

2 Using a template (explained in Part 1), add a one-third crease.

3 Rotate the double layer by 90 degrees and add a onethird crease. Repeat on the other two sides.

4 Crease and unfold two diagonals. Turn the paper over.

5 Fold from one side to the opposite side, crease, and unfold in both directions.

6 Fold all four corners to meet the nearest crease intersections.

7 This is how your paper should look. Turn the paper over.

8 Fold the raw (original) edges in to meet the one-third creases.

9 Fold all four corners to the center, crease, and unfold. Rotate the paper 90 degrees.

10 Fold in half from top to bottom.

11 Reverse the upper-yellow corners inside on the existing creases.

12 Ease out a yellow flap (actually a double flap) on either side.

13 This is how your paper should look. Repeat steps 10 through 12 on the other axis.

14 Carefully flip the model inside out—the creases

indicated stay the same, but the sides should slope down toward the center. Turn the paper over.

15 The first unit is complete. Make five more units

using whatever color combinations appeal to you.

Assembly

1 Arrange two units as indicated. Slide a yellow flap into the layers of the green flap.

2 Here is the underlying structure of assembly.

3 Add a third unit, tucking green into blue and blue into yellow.

4 Here is the underlying structure of assembly for the

half-complete model. Repeat on the other side with the remaining three units.

Assembling the First Two Units In an ideal world, you want the two layers of the yellow flap on either side of the hidden layer on the green flap. But if all you can do is push a double layer between the other double layer, it will work.

Leaved Octahedral Skeleton Design by Denver Lawson

With six units made up of three facing pairs, this design can be seen as based around the form of a cube. However, you can also visualize it as an octahedron (whose corners touch the center of each face of the cube) with “leaves” emerging, hence the name of the design. Try not to be confused by the many flaps, and see the underlying structure!

Construction

1 With the colored side up, fold in half from side to side, crease, and unfold in both directions. Turn the paper over.

2 Crease and unfold both diagonals.

3 Fold each corner to the center, crease, and unfold.

4 Fold each corner to the nearest crease intersection.

5 Refold the creases from step 3. Rotate the paper 45 degrees.

6 Fold a corner to the center, crease, and unfold. Repeat on the other three corners.

7 Fold a corner to the opposite three-quarter point (as

circled). Crease and unfold. Repeat on the other three corners.

8 Fold the left and right corners to the lower corner, forming a preliminary base.

9 Reverse the right corner inside.

10 Repeat step 9 to the other three corners.

11 Crease and unfold two diagonals in the central square.

12 Fold the lower corner to the top, reversing the sides inward.

13 Repeat step 12 on the underside.

14 Fold a flap from left to right. Repeat on the underside.

15 Repeat step 12 on both sides of the model.

16 Open all four flaps out halfway.

17 The first unit is complete. Make another five units

following the previous steps, using two sheets each of three different colors.

Having Trouble? Because the size of each completed unit is relatively small compared to the initial square, you may have some trouble seeing the structure. If that’s the case, feel free to use larger paper squares than normal.

Assembly

1 Fold one of the outer flaps of a unit in half.

2 Arrange two units with the undersides facing each

other. Insert the flaps of the green unit into the pockets on the blue unit.

3 Once inserted, lift up the green unit by 90 degrees.

4 Here’s how the two units should relate to the underlying cube geometry.

5 Lift up the central flap, so it duplicates step 1.

6 This is how your model should look so far.

7 The third unit will be added based on this underlying geometry.

8 The central green unit is inserted into the next blue unit in the same way as before.

9 This shows how the three units are interlocked.

10 Attach the remaining three units using the same

principle—two opposite flaps of each unit are tucked into other units, while two flaps accept flaps from other units.

ALPHA BOOKS Published by Penguin Random House LLC Penguin Random House LLC, 375 Hudson Street, New York, New York 10014, USA • Penguin Random House LLC (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.) • Penguin Books Ltd., 80 Strand, London WC2R 0RL, England • Penguin Ireland, 25 St. Stephen’s Green, Dublin 2, Ireland (a division of Penguin Books Ltd.) • Penguin Random House LLC (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty. Ltd.) • Penguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi—110 017, India • Penguin Random House LLC (NZ), 67 Apollo Drive, Rosedale, North Shore, Auckland 1311, New Zealand (a division of Pearson New Zealand Ltd.) • Penguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa • Penguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England Copyright © 2015 by Penguin Random House LLC

All rights reserved. No part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author’s rights. Purchase only authorized editions. No patent liability is assumed with respect to the use of the information contained herein. Although every precaution has been taken in the preparation of this book, the publisher and author assume no responsibility for errors or omissions. Neither is any liability assumed for damages resulting from the use of information contained herein. For information, address

Alpha Books, 6081 East 82nd Street, Suite 400, Indianapolis, IN 46250. IDIOT’S GUIDES and Design are trademarks of Penguin Random House LLC. eISBN: 9781615648481 Library of Congress Catalog Card Number: 2015930789 17    16    15                8     7     6     5     4     3     2     1 Interpretation of the printing code: The rightmost number of the first series of numbers is the year of the book’s printing; the rightmost number of the second series of numbers is the number of the book’s printing. For example, a printing code of 15-1 shows that the first printing occurred in 2015. Printed in China Note:       This publication contains the opinions and ideas of its author. It is intended to provide helpful and informative material on the subject matter covered. It is sold with the understanding that the author and publisher are not engaged in rendering professional services in the book. If the reader requires personal assistance or advice, a competent professional should be consulted. The author and publisher specifically disclaim any responsibility for any liability, loss, or risk, personal or otherwise, which is incurred as a consequence, directly or indirectly, of the use and application of any of the contents of this book. Most Alpha books are available at special quantity discounts for bulk purchases for sales promotions, premiums, fund-raising, or educational use. Special books, or book excerpts, can also be created to fit specific needs. For details, write: Special Markets, Alpha Books, 375 Hudson Street, New York, NY 10014. Trademarks:       All terms mentioned in this book that are known to be or are suspected of being trademarks or service marks have been appropriately capitalized. Alpha Books and Penguin Random House LLC cannot attest to

the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.