GEOMETRIC
ORIGAMI

By Faye E. Goldman
Illustrated by Marcio Noguchi

Images

Thunder Bay Press
An imprint of the Baker & Taylor Publishing Group
10350 Barnes Canyon Road, San Diego, CA 92121
www.thunderbaybooks.com

Copyright © 2014 Thunder Bay Press

Developed by The Book Shop, Ltd.

Instructions and models by Faye E. Goldman based on Snapology method by Heinz Strobl
Illustrations by Marcio Noguchi
Book designed by Tim Palin Creative

Photography by Andrew Werner Photography

Copyright under International, Pan American, and Universal Copyright Conventions. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage-and-retrieval system, without written permission from the copyright holder. Brief passages (not to exceed 1,000 words) may be quoted for reviews.

“Thunder Bay” is a registered trademark of Baker & Taylor. All rights reserved.

All notations of errors or omissions should be addressed to Thunder Bay Press, Editorial Department, at the above address. All other correspondence (author inquiries, permissions) concerning the content of this book should be addressed to The Book Shop, Ltd. New York, New York 10010.

www.thebookshopltd.com

ISBN-13: 978-1-62686-118-3
ISBN-10: 1-62686-118-8

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Acknowledgments

Thanks to Heinz Strobl for creating and sharing his technique.

He also provided some of the initial polyhedron pictures.

Thanks to Jeannine Mosely for help in visualizing the underlying mathematics of the shapes.

Thanks to Patty Grodner for diagramming the first egg.

Thanks to Arlene Gorchov for first learning the technique and passing it on to me.

Most of all, thanks to my husband Leonard, who puts up with me.

INTRODUCTION

This book contains a collection of geometric origami models made from folding strips of paper using a special technique developed by Heinz Strobl. Known as “Snapology” (you’ll understand why later), the technique employs the two most basic folds in origami—the valley fold (fold up to make a V-shape) and the mountain fold (fold behind to make a upside-down V-shape)—to build fantastic 3D models of great beauty and durability. This means that nearly every model in this book is accessible to beginning-level folders.

This is not to say that the building of these models is easy and that patience and practice isn’t required. These models are challenging, and they will require perseverence. But they are extremely rewarding and quick to build.

Most of these models have geometric polyhedra at their core—quite literally! If you recall from your high school geometry class, polyhedra (singular: polyhedron) are three-dimensional figures with many sides. Each of the models in this book is made of prisms—small triangles, squares, hexagons, etc.—that protrude from an underlying structure. Strip away the protrusions and you’ll find a tetrahedron (pyramid), a cube (hexahedron), or another such shape—polyhedra that are, to my mind, as beautiful in their own right as the paper-strip models are.

You’ll learn how to make paper-strip models based on the Platonic solids—polyhedra made of only one shape (named for the philosopher Plato), as well as the Archimedean solids—polyhedra made of two or more shapes (and named for the Greek mathematician, Archimedes). It is this connection to geometry that originally drew me to this particular type of paper folding. My background is in math and chemistry and I love making things.

BUILDING MODELS WITH PAPER STRIPS

There are two types of strips used to make these models:

•Scaffold (scaffolding) strips make up the underlying structure. They are made in the shapes of the polygons that make up the model. The length of the strips is always twice the number of sides in the polygon. To make scaffolds for triangles, for instance, use strips that are 1 × 6 because the strip is wrapped around itself twice. How long should strips for squares be? If you said 1 × 8 (two squares for each of four sides), you would be correct.

•Hinge strips connect the scaffolds. Hinges are what is most visible in the finished model. They come in two basic sizes: 1 × 4 and 1 × 6. If you are building something with a lot of prisms that are pretty tightly packed, a 1 × 4 hinge is fine. But if there aren’t a lot of prisms, or if the angles between the prisms are pretty wide (90° or larger), you need a little extra length to tuck under layers so the ends stay put. 1 × 6 strips are best in that case, but they can be a little tricky to connect. As an alternative, you can make all hinge strips 1 × 4 and glue the ends down.

You can use the same color for both the scaffolds and the hinges, but sometimes using coordinating colors looks nice, since the scaffold color always peeks through at the edges.

Scaffold and hinge strips should be folded so all the folds go in the same direction because they will be wrapped around themselves or other strips. Mountain folds are made on the outside of the strips—this is the side that is the most visible. If you make a mistake and fold valley folds, don’t despair. Just reverse the creases (fold in the other direction) to change a valley fold to a mountain fold. If the strips you are using have the same color on both sides, it is even easier to fix. Just turn the strip upside down (a valley fold is an upside-down mountain fold).

WHAT YOU’LL NEED

You can use just about any kind of paper to make these models—computer paper, colored paper, even wrapping paper—as long as it holds a fold nicely and isn’t too stiff (I don’t recommend card stock, for instance).

Gift-wrapping ribbon can also be used; just make sure that the ribbon will hold a crease when folded. Ribbon comes in such beautiful colors and prints these days, and the kind with gold edging makes great-looking models. Polypropylene ribbon works well. Fabric ribbons generally do not. Beginners should use ribbon over 3/4 inch wide. As you gain skill, you can try ribbon as narrow as 3/8 inch. Don’t go wider than 2 inches—it’s too floppy.

The Christmas season is a great time to go ribbon shopping, but don’t buy a large quantity until you can try it and be sure that it will work. When using ribbon, cut it into lengths no longer than about 60 times the width. When paper-strip ladders are made with ribbon longer than that, the ladder tends to get out of control and difficult to use.

You will also need a craft knife in order to have better control over the cuts you make on the strips. A self-healing cutting mat is also useful when using a craft knife. And when you are ready to cut strips from your own paper supply, you may want to use a paper cutter.

GETTING STARTED

•Read project instructions through before starting to build. It is always good to have an idea of the scope of any project before you get started.

•Check your paper supplies. Before beginning a project, make sure there are enough strips of the same pattern to complete the project.

•Make accurate cuts. Cut the strips into proper sizes as close to the middle of the crease as possible. If the strip is not cut in the middle, there will be an extra bit on one side, and the other side will be too short. The scaffold strips have more tolerance for error. A short hinge unit might flap up and not stay in place. Don’t worry if this happens; you can always use a drop of glue to keep the strip in place.

•Prepare your strips (see the next section)!

HOW TO PREPARE THE STRIPS

To prepare the strips for these models, you really only need to know the two most basic folds in origami—mountain folds and valley folds.

To make a mountain fold, just fold the paper over and crease. The crease (or spine) that sticks up like a mountain is the mountain crease.

A valley crease is the reverse side of a mountain crease. If you turn the mountain fold over, you will notice that the fold makes a V-shape, like a little valley. You can make a valley crease by folding the paper up.

Once you know these creases, preparing strips for these models is pretty simple. First, I will outline my preferred method, making paper-strip ladders.

1Take one strip of the scaffold paper and one strip of the hinge paper. Place one of the strips across the other and align the edges. Fold the bottom strip over the top strip.

2Fold the lower strip across the top strip.

3Rotate the strips 90° clockwise.

4Repeat steps 2 and 3 until all the paper is used up.

5Make the creases perfectly perpendicular to the edges of the paper. It will form a zigzag.

6Carefully separate the two strips.

7All of the creases should go the same way, so you will need to reverse some of the creases. The side with the mountain creases should be the side you want to show.

The strips are now ready to be cut into the lengths needed for a project. Cut as close to the middle of the crease (or spine) as possible.

Other ways to Prepare Strips

Using a Wrap-around Tool

For this method, you will make a little tool using one of the strips. Cut it to a length that is about four times the width of the strip.

1.Fold the strip in half to strengthen it.

2.Insert a strip of the length you want between the layers of the tool. Place it so the wrong side (the side you don’t want seen) is face up. Fold the strip across the tool.

3.Fold the strip behind the tool.

4.Repeat steps 2 and 3 until the end of the strip. It is OK if the end of the strip doesn’t meet the edge of the tool. The last partial square can still be used.

5.Unroll the strip and prepare for the project.

The above method can be used for scaffolding of any length, or for 1 × 6 hinges. I don’t recommend using it for 1 × 4 hinges.

Measuring

You can always simply measure your strips and mark where they should be folded. For instance, if you are making a 1 × 4 hinge strip, measure the width of the strip and multiply by four. Then pencil off marks at the ¼ mark, the ½ mark, and the ¾ mark on the wrong side of the strip (the side you don’t want seen). Make valley creases at the marks.

IMPORTANT SKILLS

MAKING UP THE SCAFFOLD UNITS

The length of the strips you prepare for each project’s scaffold units is determined by the shape of the unit. If a unit is a triangle, the strip should be 1 × 6, because scaffold units have two layers and triangles have three sides. If the unit is a pentagon, the strip should be 1 × 10, because it has five sides and is wrapped around twice.

It is a good idea to make all the scaffold units up but one before you start building, so you have everything ready at your fingertips. However, once you start building models with large numbers of prisms, you may want to make up scaffold units in batches.

Start by wrapping the strip around itself twice. Then squeeze and sharpen the edges. Now the scaffold is ready!

ATTACHING UNITS WITH 1 × 4 HINGES

1.Fold a hinge strip in half.

2.Feed one half through a scaffold unit, then fold the end down. This is known as “snapping the end down.”

3.Feed the other half of the hinge through another scaffold unit and fold the end down between the two prisms.

ATTACHING UNITS WITH 1 × 6 HINGES

Sometimes hinges need extra length so they can be tucked under another layer to hold them down, so the attaching process is a little different.

1.Reverse the crease on the ends of the 1 × 6 hinge strips so each has four valley folds and one mountain fold.

2.Position a scaffold unit so the open flap faces up.

3.Thread a hinge (the side with three valley folds) through and around the prism side. Fold up the long end.

4.Thread the other side of the hinge through another prism.

5.Fold this end down, tucking the last square under the first unit. Push the new prism up so it is perpendicular to the first.

ATTACHING THE FINAL UNIT

This part is always a little tricky because you have so little room to maneuver. Attach the final hinges to the existing model first and wrap up the last prism around the hinges.

TROUBLESHOOTING TIPS

•When cutting strips apart, try to cut down the middle of the fold so the two parts will have end squares of equal size.

•Complete rings of prisms quickly. A finished ring is more stable and won’t fall apart.

•Finish up rings before building new ones. If you see a ring with all but one of the prisms connected, attach the last unit and connect the ring first before working a different spot. Then move on to the next most complete ring.

•Discovered that you’ve use the wrong color hinge by mistake? If they aren’t glued down, they are easy to replace with the correct color.

DECORATION UNIT

This is a great first model because you get to practice making combination units with prisms and hinges, plus it gives you the opportunity to make a finished model after making just a few practice pieces. A finished Decoration Unit makes a lovely brooch when you glue a bar pin to the back and add a crystal to the center.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Fold the scaffold units in preparation for building: using a 1 × 6 strip, wrap it twice around itself to form a triangle with two layers. Squeeze and sharpen the edges. Repeat to make the other triangle units.

3Place two triangles on the work surface so the sides with the loose flaps face each other.

4Fold a 1 × 4 strip (hinge) in half. Feed one half through a triangle, then fold the end down. You have made a combination unit.

5Feed the other half of the hinge through the other triangle and fold (snap) the end down between the two triangles.

6Create another combination unit as in step 4.

7Attach the new unit to the existing model on an empty side as in step 5.

8Make two more combination units.

9Attach them to the existing model on empty sides to make a ring of five triangles.

10Use the final hinge to attach the first and last prisms.

TIP:

If the end squares of the hinges won’t stay shut, it is OK to glue them down.

ICOSAHEDRON

Once you’ve whetted your appetite with a Decoration Unit, you’ll want to move on to the next level by building a Decoration Unit out into this Icosahedron. An Icosahedron is a Platonic solid—in other words, it is made up entirely of one shape (in this case, triangles). Note that each vertex (where all angles come to a point) is a ring of five triangles.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Fold the scaffold units in preparation for building: using a 1 × 6 strip, wrap it twice around itself to form a triangle with two layers. Squeeze and sharpen the edges. Repeat to make the other triangle units.

3Make a finished Decoration Unit (see page 15).

4Make a combination unit (hinge and scaffold).

5Attach the combination unit to one of the open triangles on the existing model.

6Make two more combination units and attach them so there is another ring of five triangles adjacent to the first.

7Connect the first and last triangles of the new ring with a hinge strip.

8Using up all the remaining triangles but one, make more combination units and attach to the model in rings of five triangles. Work to complete the rings quickly to stabilize each new section as it is made.

TIP:

Fill up the most complete rings first before moving on to the next spot.

9The last triangle is the hardest to put in because there is little room to maneuver. Thread the halved hinges through the empty sides of the model first and snap the ends down.

10Stand the other ends of the hinges up. Form the last prism by wrapping the scaffold around the hinges twice.

11Fold the hinge ends down and snap them into place.

TETRAHEDRON

Because the angles between these prisms are so wide, we need to use a different kind of hinge. It is longer and the additional squares on the end of each will be used to tuck into an extra layer to hold the unit in place. In this model, all vertices consist of rings of three triangles.

TIP:

Make the scaffolds a different color than the hinges; the new color will peek through at the edges.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the four scaffold units in preparation for building.

3Reverse the crease on the ends of the six hinge strips so each has four valley folds and one mountain fold.

4Position a triangle so the open flap faces up. Thread a hinge (the end with three valley folds) through the triangle.

5Wrap the hinge end around the triangle and tuck under the edge.

6Thread the other side of the hinge through another triangle.

7Push the triangle up so it stands at a right angle to the first triangle.

8Fold this end down, tucking the last square under the first unit.

9Make a combination unit (one hinge snapped on to one scaffold). Attach it to the model.

10Connect the first and last unit of this ring as in steps 5 and 8.

11Turn the model over.

12Thread the last three hinges through the empty sides of the triangles. 12 Wrap the short ends around first and tuck the end squares under each unit.

13Straighten the long ends so they stand up.

14Wrap the last scaffold strip twice around the upright hinges.

15Fold the hinge ends down and tuck the end squares through the adjacent units.

OCTAHEDRON

The angle between prisms isn’t as wide as with the Tetrahedron, but the 1 × 6 hinges still work better for this model.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold units in preparation for building.

3Reverse the crease on the ends of the hinge strips so each has four valley folds and one mountain fold.

4Make a Tetrahedron through step 9.

5Make a combination unit (scaffold plus hinge) and add it to the model to make a ring of four triangles.

6Connect the first and last unit with a hinge.

7Turn the model over.

8Continue adding combination units to make another ring of four. Connect the first and last prism with a hinge unit.

9Add the final ring: attach the first three prisms and connect free sides when possible.

10Thread the final three hinges through the empty prism sides and stand the other ends up.

11Wrap the last scaffold strip around the hinge ends.

12Fold the hinge ends down and tuck under layers of the adjacent triangle units.

CUBE (HEXAHEDRON)

Hexahedrons—otherwise known as cubes—are the only Platonic solids to be made entirely of squares. Like the Tetrahedron, it has vertices that are rings of three, but the shapes are squares instead of triangles. Since the angles between connected prisms are so wide, use the longer (1 × 6) hinges here.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold squares in preparation for building.

3Reverse the crease on the ends of the 12 hinge strips so each has four valley folds and one mountain fold.

4Make a combination unit from a square and a hinge (see Tetrahedron/Steps 4–5, page 23).

5Thread the other end of the hinge through another square. Wrap the hinge end around and tuck it into the first square.

6Make another combination unit. Attach it to the existing model. Use an extra hinge to attach another side where it butts up against another prism. This makes a ring of three prisms.

7Attach yet another combination unit. Use another hinge to attach another empty side of the new unit.

8Turn the model over.

9Attach another combination unit and use two additional hinges to complete both rings of prisms of which the combo unit is now part.

10Thread the last hinge strips through the empty squares and snap ends in place.

11Stand the other hinge ends up. Wrap the last scaffold strip twice around the hinges.

12Fold the hinge ends down and tuck under the adjacent square units.

DODECAHEDRON

This is the only Platonic solid that is made entirely of pentagons. It has 20 vertices, each formed by the meeting of three pentagons, and 12 edges. Because the prisms are so close together, you can use 1 × 4 hinges for this model.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold pentagons in preparation for building.

3Make a combination unit from a pentagon and a hinge (see Decoration Unit/ Step 4, page 15).

4Thread the other end of the hinge through another pentagon. Snap the end into place between the two pentagons. Push the pentagon up so it stands at a closer angle to the first one.

5Make another combination unit.

6Attach the new unit to the existing model and snap the hinge end into place.

7Use an extra hinge to attach another side where the new pentagon butts up against another prism.

8Use hinges and all scaffolds but one to make remaining combination units.

9Keep attaching combination units to make rings of three pentagons. Use extra hinges to attach empty sides to the model where possible.

10To place the final pentagon, thread the last five hinges through the empty sides on the model and snap ends down.

11Stand the other hinge ends up. Wrap the last scaffold around them twice, then snap the hinges in place between the prisms.

TRUNCATED TETRAHEDRON

The Truncated Tetrahedron is the shape that is formed after the points of a tetrahedron are cut off. What remains are four hexagonal faces and four triangular faces. You will need to use the 1 × 6 hinges to connect the hexagons to each other, but you can substitute 1 × 4 hinges when connecting hexagons and triangles, if desired.

The model shown has 12 triangle-hexagon hinges in one color and 6 hexagon–hexagon hinges in a second color.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold hexagons and triangles in preparation for building. Prefold the 1 × 6 hinge strips.

3Connect two hexagons with a hinge strip (follow instructions for using 1 × 6 hinges to connect prisms on page 12).

4Make a combination unit from a triangle and a hinge strip (see Tetrahedron/Steps 4–5, page 23).

5Attach the combination unit to an empty side between the connected hexagons.

6Use another hinge to attach the other sides.

7Repeat steps 4 through 6 on the opposite side of the connected hexagons.

8Make a combination unit of a hexagon and a hinge strip.

9Attach the combination unit to an edge of a hexagon next to a triangle.

10Attach other free edges (between hexagon and triangle and hexagon and hexagon) with hinge strips.

11Use hinge strips and all the remaining scaffold units but one to make more combination units.

12Continue adding combination units to free sides, keeping the following rule in mind: each vertex (point) is the meeting point of three prisms (a hexagon, a triangle, and a hexagon). Use additional hinge strips to attach free sides when required.

13To make the last prism, thread the hinge strips through the empty sides first and tuck ends under.

14Straighten out the other hinge ends and wrap the last scaffold strip around them twice.

15Tuck ends away.

CUBOCTAHEDRON

Like its name suggests, the Cuboctahedron resembles both a cube and an octahedron. It consists of six squares and eight triangles. Note that each vertex consists of a ring of four shapes: triangle, square, triangle, square.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold squares and triangles in preparation for building.

3Make a combination unit from a triangle and a hinge strip (see page 15, step 4).

4Attach the combination unit to a square (see “Attaching Units with 1 × 4 Hinges,” page 11).

5Repeat steps 3 and 4 on an adjacent edge of the square.

6Make a combination unit of a square and a hinge strip.

7Connect the square combo unit opposite the first square. Use an extra hinge to connect the free side.

8Use hinge strips and all the remaining scaffold units but one to make more combo units.

9Continue attaching combination units to the existing model, keeping this rule in mind: each vertex (point) should join a triangle, a square, a triangle, and a square. Attach edges with extra hinge strips when possible.

10To place the last scaffold unit, thread the hinge strips through the empty sides first and snap ends down.

11Stand the other hinge ends up and wrap the scaffold strip around them twice.

12Snap ends down.

ICOSIDODECAHEDRON

Made of 12 pentagons and 20 triangles, this solid has two of each kind alternating around each vertex. Notice how the figure looks rounder the more faces it has. Since the prisms fit so tightly together, you can use 1 × 4 hinges for this project.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold pentagons and triangles in preparation for building.

3Make a combination unit from a triangle and a hinge strip.

4Attach the combo unit to a pentagon (see “Attaching Units with 1 × 4 Hinges,” page 11).

5Repeat steps 3 and 4 on an adjacent edge of the pentagon.

6Make a combination unit from a pentagon and a hinge unit.

7Attach it to the model opposite the other pentagon. Use another hinge strip to attach the free edges.

8Make a pentagon combo unit.

9Attach it to one of the the triangles.

10Make a triangle combo unit.

11Attach it to the model between the pentagons. Use a hinge strip to attach the free edges.

12Make another triangle combo unit.

13Attach it to the model between the pentagons. Use a hinge strip to attach the free edges.

14Use hinge strips and all remaining scaffold units but one to make combo units.

15Continue adding combo units to the existing model. Keep in mind that each vertex has a triangle, a pentagon, a triangle, and a pentagon. Use hinge strips to connect free edges whenever possible.

16To attach the final prism, thread the hinge strips through the empty sides first and snap ends down.

17To attach the final prism, thread the hinge strips through the empty sides first and snap ends down.

RHOMBIC TRIACONTAHEDRON

A rhombus has four equal sides like a square and its opposite sides are parallel, but its angles aren’t equal. In other words, it makes a diamond shape. This solid is made up entirely of rhombi, but the vertices have different numbers of them. Twenty vertices have three rhombi and 12 vertices have five rhombi. You can use 1 × 4 hinges for this model.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold squares in preparation for building. Don’t worry about shaping them like diamonds; when attached in the model, the prisms will naturally fall into the correct shape.

3Make a combination unit from a square and a hinge strip.

4Attach the combo unit to another square (see “Attaching Units with 1 × 4 Hinges,” page 11).

5Repeat steps 3 and 4 on an adjacent edge of the model. This makes a vertex with three rhombi. Note that the angles that meet are all obtuse.

6Start an adjacent vertex that will have five rhombi: make a combination unit from another square and a hinge unit.

7Attach it to the model.

8Make another combo unit.

9Attach it to a free side as shown.

10Make another combo unit.

11Attach it to the model between the newest squares. Use an extra hinge strip to attach the free edges of the adjacent rhombi. The adjacent vertex with five rhombi is complete. Note that the angles that meet in rings of five are all acute.

12Make another combo unit.

13Attach it to the model between the rhombi as shown. Use a hinge strip to attach the free edges.

14Repeat step 12 and 13 to complete rings of three rhombi around the ring of five.

15Make combo units from the hinge strips and all remaining rhombi but one.

16Continue adding combo units to the existing model in either rings of three rhombi or in rings of five. You will know how many to add because the number always alternates on adjacent vertices. Use extra hinge strips to attach free edges when possible.

17To attach the final prism, thread hinge strips through the empty sides first and snap ends down.

18Stand the loose ends up and wrap the final scaffold strip around them twice. Snap ends down.

SNUB DODECAHEDRON

There are 12 pentagons and 80 triangles in a snub dodecahedron—that’s a lot of prisms to make! Each vertex is formed by a pentagon and four triangles. When you are working on projects with this many pieces, you may want to consider making the prisms up in batches. You can use 1 × 4 hinges for this model.

The sample shown uses 12 pentagons made in six different colors, surrounded by triangles of a seventh color.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold pieces in preparation for building.

3Make a combination unit from a triangle and a hinge strip (see Decoration Unit/Step 4, page 15).

4Attach the combo unit to a pentagon that is the same color as the hinge (see “Attaching Units with 1 × 4 Hinges,” page 11). In this color scheme, the pentagon hinges always match the scaffold color.

5Repeat steps 3 and 4 on an adjacent edge of the pentagon.

6Make two more triangle combo units and attach them to the model between the two triangles. These triangle combo units—ones that do not share edges with the pentagons—are always made of green background color. Use extra hinges to connect free edges when possible. This completes one vertex.

7Make three more triangle combo units and attach them as shown. Use extra hinges to connect the free sides.

8Repeat step 7 on the next empty edge of the pentagon.

9Repeat step 7 on the last empty edge of the pentagon.

10Make two more triangle combo units and attach them to the model, completing the ring of triangles around the pentagon. Use an extra hinge to connect the free sides.

11Make a combination unit from a pentagon and a hinge strip that matches the pentagon.

12Attach the pentagon combo unit to one of the triangles.

13Make a triangle combo unit from the background color.

14Place it so the point touches the point of the pentagon. Attach it to the free edge of the adjacent triangle.

15Make another triangle combo unit and attach it between the last triangle and the pentagon.

16Use an extra hinge to connect free sides. Notice that you have completed another vertex: four triangles and a pentagon.

17Make a matching hinge-pentagon combo unit and attach it to the next adjacent group of four triangles, completing that vertex.

18Use hinge strips and all scaffold pieces but one to make remaining combo units.

19Continue adding combo units to the model, using extra hinges to connect free sides when possible. Keep in mind that you are creating a pattern of colored pentagons separated from each other by rows of green triangles.

20To attach the last prism, thread the hinge strips through the empty sides first and snap the ends down.

21Stand the loose ends up and wrap the final scaffold strip around them twice. Snap ends down.

ZONOHEDRON

A zonohedron is a polyhedron with faces that are all centrally symmetric. It can have different numbers of sides; the model shown here has 90 and is made entirely of rhombi, so it is also a rhombic enneacontahedron (enneacontahedron means 90 sided). It has vertices of three, five, or six rhombi, and as with earlier models, the rhombi change shape according to the number connected at each vertex. Since the prisms fit so tightly together, you can use 1 × 4 hinges for this project.

To match this sample, use a different hinge color to make the center of each ring of five rhombi. There are 12 rings of five, but only six colors. Locate the matching colors opposite each other. The other hinges are made from the same background purple. The scaffolds are all made from the same background color.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold pieces in preparation for building.

3Make a Rhombic Triacontahedron (see page 50) through step 14.

4Make three rhombus combination units (rhombus plus a hinge) from the background color.

5Look closely at the vertex indicated. Notice that all three of the connected angles are acute (less than 90°). Attach the combination units here to make a ring of six rhombi. Use an extra hinge to attach the first and last rhombi.

6Make 12 more rhombus combination units. Hinge colors will change based on location.

7Repeat step 5: attach the combination units to the other vertices of three acute angles to make additional rings of six rhombi.

8Notice the new pattern outlined: you can see two adjacent rings of six rhombi that share one skinny rhombus between them.

9Make six more rhombus combination units.

10Locate the vertex of two rhombi on the lower right edge. Attach the combination units to this vertex to make a ring of five rhombi.

11This is the basic building tile for this particular zonohedron. Notice that you can now see three completed rings of six rhombi. The skinny rhombi that connect the rings of six are shapes that separate three completed rings of five rhombi.

12Make all remaining rhombus combination units but one.

13Continue to attach the units to the existing model to make rings of five or six rhombi. There will be 12 vertices with rings of five matching prisms. Keep these rules in mind: the rings of six share a skinny rhombus with each other (notice that the rings show alternating regular and skinny rhombi) while the rings of five do not share any sides with each other. The rings of five and the rings of six are connected to each other at vertices of three angles.

14When you are ready to attach the final prism, attach the hinges first: thread the hinge strips through the free sides and snap the ends down.

15Stand the other ends up and wrap the scaffold strip around them twice. Snap ends into place.

EGG

This is the first model in this collection that has a more limited symmetry—it is longer than it is wide. If you study it carefully, you will notice that it can be divided into bands. Around the top and bottom squares there are bands of hexagons, while the band around the middle is made up of eight hexagons connected to eight pentagons.

To match the model shown here, use hinges of three different colors: use one color for the inside of the wheels, another color to connect wheels to pentagons or squares, and a third color to connect wheels to wheels. The scaffolds are all made of the same background color.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold pieces in preparation for building. You will need two squares plus 16 wheels made of six triangles in addition to eight pentagons.

3Make a Decoration unit (see page 15) but use six triangles instead of five. This is your first wheel. Make 15 more.

4Make a combo unit from a hinge and a square. Attach it to a wheel.

5Attach wheels to the remaining sides of the square with hinges. Then use extra hinges to attach the wheels to each other.

6Attach a pentagon unit to the model between two adjacent wheels with two hinge strips, one for each wheel.

7Attach three more pentagon units between the other wheels in the ring.

8Attach a wheel between each pentagon, using three hinges to connect the sides.

9Repeat with three more wheel units.

10Attach another band of wheels between the first band of wheels. Use hinge strips to connect the sides between the wheels, and between the wheels and the pentagons.

11Attach the last four pentagons to the model between the wheels, using hinge strips to connect the free sides.

12Attach the last four wheels between the pentagons, using hinge strips to connect the sides between the wheels, and between the wheels and the pentagons.

13To place the final prism (the remaining square), thread the remaining hinge strips through the free sides and snap ends down.

14Stand the other hinge ends up and wrap the scaffold strip around them twice. Snap ends down.

BRACELET

In this model, a long strip of interconnected triangles, two or three rows deep, is linked to make a bangle. Look for the thinner folding strips to make this model. I also like to use pretty 3/8-inch ribbon for this one, maybe something with gold edging. Ribbon comes is all kinds of colors and you can fold it into paper-strip ladders quite easily. I first cut the ribbon into long strips of about 24 inches before making the paper-strip ladders.

1Prepare the strips (of paper or ribbon) using your choice of methods and cut the quantity of hinges and scaffolds indicated below. Your wrist may be larger or smaller, so be sure to measure for your size.

2Make the scaffold pieces in preparation for building. Because you are making something that fits your wrist, you may need more or less than the number indicated here. But plan to make at least 72 triangle units and 90 hinges to start.

3Make a Decoration Unit (see page 15) but use six triangles instead of five to make a wheel.

4Make a combination unit from a triangle and a hinge strip.

5Attach the combination unit to one of the wheel’s free sides.

6Make another triangle combination unit and attach it to the wheel on the free side in the row below. The point of the triangle will meet the point of the combination unit you added in step 5. Remember, units for the bottom row are one color, and units for the top row are a different color.

7Make two more triangle combination units, one in each color. Hinge colors will change based on location.

8Attach these next to the units you just added. Note that you will need an extra hinge to connect them to each other. Alternate colors for these.

9Continue making more combination units. Repeat steps 4 through 8 to attach them to the model until it can be wrapped comfortably around your wrist.

10If you like, you can make the band thicker at this point. Add another row of triangle units to the entire length of the band.

11Once the desired length is reached, complete the bracelet: connect the first and last columns with hinges.

BUCKYBALLS

Buckyballs, named for architect/inventor Buckminster Fuller, are so called for their resemblance to Fuller’s geodesic dome structures. Buckyballs can be of different sizes and shapes, but they are always made only of pentagons and hexagons—each formed from equilateral triangles—and there are always 12 places where the triangles form pentagons. You may use 1 × 10 scaffold strips to make these pentagons, if desired.

To match the model shown here, use the same color—green—hinges for the centers of the rings of five, and a second color—orange—for all other hinges.

1Prepare the strips using your choice of methods and cut the quantity of hinges and scaffolds indicated below.

2Make the scaffold pieces in preparation for building. Since there are so many, you can make them up in batches.

3Make a Decoration Unit (see page 15).

4Make four triangle combination units. Two will have orange hinges and two will have green.

5Add the combination units to the existing model at an open vertex to make a ring of six triangles. Use an extra hinge to connect the first and last triangle in the ring.

6Make three more triangle combination units.

7Use these units to create a ring of six triangles around an adjacent open vertex. These will have orange hinges. Use an extra hinge to connect the first and last triangle in the ring.

8Make three more triangle combination units.

9Use these units to create a ring of five triangles extending from the bottom of the model. Use an extra hinge to connect the first and last triangle.

10Repeat steps 8 and 9 to make another ring of five triangles on the other side of the model.

11Make another triangle combo unit and use it to fill in the empty space between the rings of five. Note that it will complete a ring of six triangles.

12This is a completed tile, the basic building block of this particular Buckyball. Note that it has three vertices and has a vaguely triangular shape.

13Continue building up the model in this way, starting a new tile with a ring of six triangles at either side of the previous tile’s base. No matter what the size and shape of a Buckyball, it will always have 20 of these tiles.

14As with other models, the placement of the final prism is tricky. Thread the hinge strips through the free sides first and snap ends down. Stand the other ends up and wrap the last scaffold around them twice. Snap ends down.

Bibliography

Mathematics

Coxeter, H.S.M. Regular Polytopes. Mineola: Dover Publications, 1973. Print.

Cromwell, Peter R. Polyhedra. Cambridge: Cambridge University Press, 1997. Print.

Cundy, H.M. and Rollett, A. P. Mathematical Models. Oxford: Oxford University Press, 1961. Print.

Holden, Alan. Shapes, Space and Symmetry. Mineola: Dover Publications, 1991. Print.

Sutton, Daud. Platonic & Archimedean Solids. New York: Walker & Company, 2010. Print.

Wenninger, Magnus J. Polyhedron Models. Cambridge: Cambridge University Press, 1971. Print.

Eric W. Weisstein, MathWorld A Wolfram Web Research. Available:

http://Mathworld.wolfram.com/ArchimedeanSolid.html

http://Mathworld.wolfram.com/Polyhedron.html

http://Mathworld.wolfram.com/JohnsonSolid.html

Origami

Franco, Betsy. Unfolding Mathematics with Unit Origami. Emeryville: Key Curriculum Press, 1999. Print.

Fuse, Tomoko. Unit Origami: Multidimensional Transformations. New York: Japan Publications, 1990. Print. Kasahara, Kunihiko and Takahama, Toshie. Origami for the Connoisseur. New York: Japan Publications, 1987. Print.

Kawamura, Miyuki. Polyhedron Origami for Beginners. Tokyo: Japan Publications, 2001. Print.

Hull, Thomas. Project Origami. Wellesley: A.K. Peters Ltd, 2006. Print.

Montroll, John. A Constellation of Origami Polyhedra. Mineola: Dover Publications, 2004. Print.

—Origami Polyhedra Design. Wellesley: A.K. Peters Ltd., 2009. Print.

—A Plethora of Polyhedra in Origami. Mineola: Dover Publications, 2002. Print.

Mukerji, Meenakshi. Marvelous Modular. Wellesley: A.K. Peters Ltd, 2007. Print.

Simon, Lewis, Arnstein, Bennett and Gurkewitz, Rona. Modular Origami Polyhedra. Mineola: Dover Publications, 1999. Print.

Paula’s Orihouse, http://www.orihouse.com/knotology.html

Strobl, Heinz. My Strip Tease Gallery, http://www.knotology.eu/index_en.html

Organizations

Origami-USA: www.OrigamiUSA.org

British Origami Society: www.britishorigami.info