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Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality

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Explore the link between paperfolding and mathematics with this unique, well-illustrated guide to creating a world of multifaceted wonders that draws on elements of crystallography. Photographs, diagrams, and charts show how to apply mathematical principles to the origami waterbomb base to create geodesic forms. Detailed instructions, clear diagrams.

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Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality

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Overview

Explore the link between paperfolding and mathematics with this unique, well-illustrated guide to creating a world of multifaceted wonders that draws on elements of crystallography. Photographs, diagrams, and charts show how to apply mathematical principles to the origami waterbomb base to create geodesic forms. Detailed instructions, clear diagrams.

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Product Details

  • ISBN-13: 9780486423173
  • Publisher: Dover Publications
  • Publication date: 3/3/2003
  • Series: Dover Origami Papercraft Series
  • Pages: 80
  • Sales rank: 731,093
  • Product dimensions: 8.33 (w) x 10.98 (h) x 0.26 (d)

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Multimodular Origami Polyhedra

Archimedeans, Buckyballs, and Duality


By Rona Gurkewitz, Bennett Arnstein, Bill Quinnell

Dover Publications, Inc.

Copyright © 2003 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-13677-6



CHAPTER 1

I. BACKGROUND MATERIAL

a. ARCHIMEDEAN SOLIDS

b. DUALITY

c. GYROSCOPED ARCHIMEDEANS

d. GYROSCOPE MODULES

e. BUCKYBALLS

f. THE EGGS


A. ARCHIMEDEAN SOLIDS

Our first group of models is based on pairs of polyhedra, specifically Archimedean solids and their dual Catalans. The Archimedean solids were known as early as 400 B.C., although they were named at a later date after their rediscovery by Kepler and others. They are also called semiregular solids. They are the only convex polyhedra other than the Platonic solids with regular polygons for faces and with vertices that all have the same sequence of polygons meeting around them. The Platonic solids each have only one type of face; the Archimedean solids each have two or three types of faces.

The Archimedeans are related to the larger set of Johnson solids, which have regular polygons as faces but more than one type of vertex, as well as to the set of uniform polyhedra which have all vertices alike and all faces alike but not necessarily regular, and which are not necessarily convex. The Archimedeans are convex. We consider all thirteen of the Archimedean solids in this book.


B. DUALITY

Other properties of the thirteen Archimedean solids that we found useful for designing models relate to their associated spheres. All vertices of an Archimedean solid lie on a sphere, called the circumsphere, and their edges are tangent to a midsphere at the midpoints of the edges. These properties lead naturally to the standard dual, a polyhedron whose edges are perpendicular to both the Archimedean edges and to the segments from the center of the polyhedron to the midpoints of the Archimedean edges. The set of thirteen such duals is historically designated the Catalan solids. Perhaps the most striking feature of a Catalan solid is that it is composed of just one kind of face that is not a regular polygon. A face of a Catalan solid can be simply constructed by the Dorman-Luke construction, which is based on a dualization process known as polar reciprocation, with respect to the midsphere. Although topologically equivalent to the Catalan solid, a proportionately different (non-standard) dual occurs when we replace each Archimedean face center with a vertex and each Archimedean vertex with a face. Such an interchange of parts produces a dual with the same number of edges as the Archimedean, and the new faces remain all of one kind.


C. GYROSCOPED Archimedeans

Each polyhedron in our first group of models has been produced from an Archimedean through the gyroscoping process and is a melding of that Archimedean and its dual Catalan. We have chosen to designate these new gyroscoped polyhedra with the names of the Archimedeans they are derived from, although they could just as easily have been named after the dual Catalans; they have properties derived from each.

Specifically, these gyroscoped Archimedeans are made up of gyroscope modules that correspond in type to the faces of the Archimedean, namely, triangular, square, pentagonal, hexagonal, octagonal, or decagonal (see modules section). The gyroscope modules function as waterbomb bases with tabs and pockets to lock them together, so the models are not convex. However, amazingly, the moutainfolded edges of the new polyhedra correspond to the edges of the Catalans they are based on: we can see duality happening. Each face of an Archimedean has a vertex placed on it by placing a gyroscope module on it with the moun-tainfolds of the gyroscope module going from its vertex to the midpoints of the edges of the face of the Archimedean, thereby interchanging the face of the Archimedean with the vertex of the gyroscope module, on the gyroscoped Archimedean.


D. GYROSCOPE MODULES

A gyroscope module is placed on the face of an Archimedean in such a way that it is not apparent at first where the original face is. The mountain folds going from the vertex of the gyroscope module to the tabs and pockets of the module meet the edge of the pocket at the midpoint of an edge of a face of the underlying Archimedean. That is, the edges of the pockets of the gyroscope module lies along the edges of the faces of the Archimedean, but the pockets are shorter than the edges of the faces. Because of this, there are open spaces on some of the models.

As for the faces of the Catalan, they are formed by the mountain folds of the gyroscope module because of certain of their properties. Two gyroscope modules interlock with a tab from one module and a pocket from the other module. The edges of the pockets from both modules meet, so that they are perpendicular to the mountain folds that go through them in a straight line. This property further shows how the models are related to a duality process because the edges of an Archimedean are perpendicular to the edges of its dual. Also, the edges of an Archimedean are bisected by the edges of its dual, and so the mountain folds of the gyroscope modules on an Archimedean bisect the edges of the Archimedean.


E. BUCKYBALLS

The second group of models we have designed includes the buckyballs, otherwise known as the fullerenes, as well as the hypothetical buckyballs. The third group comprises the gyroscoped forms of the buckyballs and the hypothetical buckyballs (pp. 24-38)

Buckminsterfullerene, or C60, is a carbon substance discovered in 1985 by Robert F. Curl, Jr. and Richard E. Smalley of Rice University, and Harold W Kroto of the University of Sussex, England, who received the Nobel Prize for Chemistry in 1996 for this work. It was named after the designer/philosopher R. Buckminster Fuller because its molecular structure, a truncated icosahedron (like the surface pattern of a soccer ball), resembles the geodesic domes designed by Fuller. Previously, graphite and diamonds were the only known pure carbon substances (and their molecular structure is much simpler geometrically).

At the time of their discovery it was felt that buckyballs held much promise for practical applications, and some progress toward such applications has been made. Possibilities are thought to include drug delivery systems, HIV-blocking drugs, better protective coatings, and improved carbon dating. Recently, other cased structures, of silicon, have been discovered and have raised similar hopes.

The fifteen buckyballs and hypothetical buckyballs we construct are made of pentagons and hexagons, squares and hexagons, and triangles and hexagons. These are all regular polygons, so we were able to use the gyroscoping process on them, revealing a dual in the mountain folds, thus creating another group of models. We show a process for constructing each family of buckyballs and hypothetical buckyballs. Only the 3-1-1, 4-1-1, and 5-1-1 are regular polyhedra; the others are approximations.


F. THE EGGS

The last models we present are the "egg" and the "gyroscoped egg." The egg is unusual because it has an elliptical rather than spherical shape. The gyroscoped egg truly looks like a decorated holiday egg. The egg is a truncated hexadecahedron. It has forty-eight vertices and so can be made from forty-eight triangle gyroscope modules. It consists of two square rings on its ends, each surrounded by four hexagonal rings, which are themselves surrounded by eight more hexagonal rings and eight pentagonal rings (p. 8). Note that the egg is not a Johnson solid, so it does not have exactly regular faces.

CHAPTER 2

II. GALLERY OF POLYHEDRA

a. THE EGG AND THE GYROSCOPED EGG

b. ARCHIMEDEANS, CATALANS, AND GYROSCOPED ARCHIMEDEANS

c. SEEDS FOR GROWING BUCKYBALLS, BUCKYBALLS, HYPOTHETICAL BUCKYBALLS, AND THEIR GYROSCOPED FORMS


A. THE EGG AND THE GYROSCOPED EGG

The Egg
(Truncated
Hexadecahedron)

48 triangle modules arranged in:

2 square rings
8 pentagon rings
16 hexagon rings

Gyroscoped Egg

2 square gyroscope modules
8 pentagon gyroscope modules
16 hexagon gyroscope modules


B. ARCHIMEDEANS, CATALANS, AND

GYROSCOPED ARCHIMEDEANS

1. Gyroscoped Truncated Tetrahedron

Truncated Tetrahedron

Triakis-tetrahedron

Gyroscope
Modules:
4 triangles and
4 hexagons


2. Gyroscoped Cuboctahedron

Cuboctahedron

Rhombic Dodecahedron

Gyroscope
Modules:
8 triangles and
6 squares


3. Gyroscoped Truncated Octahedron

Truncated Octahedron

Tetrakis hexahedron

Gyroscope
Modules:
6 squares and
8 hexagons


4. Gyroscoped Truncated Cube

Truncated Cube

Triakisoctahedron

Gyroscope
Modules:
8 triangles and
6 octagons


5. Gyroscoped Rhombicuboctahedron

Rhombicuboctahedron

Trapezoidal Icositetrahedron

Gyroscope
Modules:
8 triangles and
18 squares


6. Gyroscoped Truncated Cuboctahedron

Truncated Cuboctahedron

Disdyakis Dodecahedron

Gyroscope
Modules:
12 squares,
8 hexagons, and
6 octagons


7. Gyroscoped Snub Cube

Snub Cube

Pentagonal Icositetrahedron

Gyroscope
Modules:
32 triangles and
6 squares


8. Gyroscoped Icosidodecahedron

Icosidodecahedron

Rhombic Triacontahedron

Gyroscope
Modules:
20 triangles
12 pentagons


9. Gyroscoped Truncated Icosahedron

Truncated Icosahedron

Pentakis-dodecahedron

Gyroscope
Modules:
12 pentagons and
20 hexagons


10. Gyroscoped Truncated Dodecahedron

Truncated Dodecahedron

Triakis-Icosahedron

Gyroscope
Modules:
20 triangles and
12 decagons


11. Gyroscoped Rhombicosidodecahedron

Rhombicosidodecahedron

Trapezoidal Hexecontahedron

Gyroscope
Modules:
20 triangles,
30 squares, and
12 pentagons


12. Gyroscoped Truncated Icosidodecahedron

Truncated Icosidodecahedron

Disdyakis Triacontahedron

Gyroscope
Modules
30 squares,
20 hexagons, and
12 decagons


13. Gyroscoped Snub Dodecahedron

Snub Dodecahedron

Pentagonal Hexecontahedron

Gyroscope Modules: 80 triangles and 12 pentagons


C. SEEDS FOR GROWING BUCKYBALLS, BUCKYBALLS,

HYPOTHETICAL BUCKYBALLS, AND THEIR GYROSCOPED FORMS


1. 3-1-1

Net

3-1-1 Truncated Tetrahedron

3-1-1 Gyroscoped Truncated Tetrahedron

Ring of one hexagon with each triangle touching one edge of hexagon

Modules:
12 triangles or
4 triangles and
4 hexagons


2. 3-3-2

Net

3-3-2 Hypothetical Bucky

3-3-2 Gyroscoped

Ring of three hexagons with triangle touching edge of two hexagons in a ring

Modules
16 triangles or
4 triangles and
6 hexagons


3. 3-3-1

Net

3-3-1 Hypothetical Bucky

3-3-1 Gyroscoped

Ring of three hexagons with triangle touching edge of one hexagon in a ring

Modules:
28 triangles or
4 triangles and
12 hexagons


4. 3-7-2

Net

3-7-2 Hypothetical Bucky

3-7-2 Gyroscoped

Ring of seven hexagons with triangle touching one edge of two hexagons in a ring

Modules:
36 triangles or
4 triangles and
16 hexagons


5. 3-7-1

Net

3-7-1 Hypothetical Bucky

3-7-1 Gyroscoped

Ring of seven hexagons with triangle touching edge of one hexagon in a ring

Modules:
48 triangles or
4 triangles
22 hexagons


6. 4-1-1

Net

4-1-1 Truncated Octahedron

4-1-1 Gyroscoped Truncated Octahedron

Ring of one hexagon with square touching one edge of one hexagon in ring

Modules:
24 triangles or
6 squares and
8 hexagons


7. 4-3-2

Net

4-3-2 Hypothetical Bucky

4-3-2 Gyroscoped

Ring of three hexagons with square touching one edge of two hexagons in a ring

Modules:
32 triangles or
6 squares and
12 hexagons


8. 4-3-1

Net

4-3-1 Hypothetical Bucky

4-3-1 Gyroscoped

Ring of three hexagons with square touching one edge of one hexagon in ring

Modules:
56 triangles or
6 squares and
24 hexagons


9. 4-7-2

Net

4-7-2 Hypothetical Bucky

4-7-2 Gyroscoped

Ring of seven hexagons with square touching edge of two hexagons in a ring

Modules:
72 triangles or
6 squares and
32 hexagons


10. 4-7-1

Net

4-7-1 Hypothetical Bucky

4-7-1 Gyroscoped

Ring of seven hexagons with square touching edge of one hexagon in a ring

Modules:
96 triangles or
6 squares
44 hexagons


11. 5-1-1

Net

5-1-1 C60 Truncated Icosahedron

5-1-1 Gyroscoped C60

Ring of one hexagon with pentagon touching one edge of one hexagon in ring

Modules:
60 triangles or
12 pentagons and
20 hexagons


12. 5-3-2

Net

5-3-2 C80 Bucky

5-3-2 Gyroscoped C80 Bucky

Ring of three hexagons with pentagon touching two edges of one hexagon in ring

Modules:
80 triangles or
12 pentagons and
30 hexagons


13. 5-3-1

Net

5-3-1 C140 Bucky

5-3-1 Gyroscoped C140 Bucky

Ring of three hexagons with pentagon touching one edge of one hexagon in ring

Modules:
140 triangles or
12 pentagons and
60 hexagons


14. 5-7-2

Net

5-7-2 C180 Bucky

5-7-2 C180 Gyroscoped

Ring of seven hexagons with pentagon touching edge of two hexagons in a ring

Modules:
180 triangles or
12 pentagons and
80 hexagons


15. 5-7-1

Net

5-7-1 C240 Bucky

5-7-1 Gyroscoped C240 Bucky

Ring of seven hexagons with pentagon touching edge of one hexagon in a ring

Modules:
240 triangles or
12 pentagons and
110 hexagons

CHAPTER 3

III. MODEL CONSTRUCTION

a. MODEL SELECTION

b. MODULE SELECTION

c. MODULE SIZING

d. NETS AS GUIDES; NAMING SYSTEM FOR POLYHEDRA

e. MODEL ASSEMBLY

f. FOUR-SIDED DISPLAY STAND FROM A PENTAGON WATERBOMB BASE


A. MODEL SELECTION

When selecting a model to make, you should first consider what is involved. In the Gallery (p. 7) you will find photos of the models, with information on the number and type of modules needed for each. For models made from more than one type of module, you will often need a different size paper for each type of module. This is to allow the modules of different types to fit together. In a following section, we give some sizings for each model's modules. These sizings produce manageably sized models with modules that are not too hard to fold. Some of the larger buckyballs and their gyroscoped forms as well as the gyroscoped Archimedeans that contain octagons and decagons are larger and require some glue.


B. MODULE SELECTION

When making models with only pentagons and hexagons, the pentagon and hexagon expanded spike ball modules can be substituted for pentagon and hexagon gyroscope modules. We include them because they have better locks, using connectors. As you might guess, these modules involve more folding and must be made large so that the connector will be a manageable size.


C. MODULE SIZING

Sizing the modules is the key to making the models. In this section our first goal is to present some sizing sets that will allow you to make models which are not too big and yet use modules that are a convenient size that are not too hard to fold. A sizing is based on a part of the starting polygon or square for the module. Different construction methods for polygons may make certain ways of sizing easier than others. Our determination of the sizing sets is based on an empirical and pragmatic method akin to what scientists did with crystals a long time ago. Basically, we folded modules and partial models and measured them, or the holes where modules need to fit.

First, consider how to make the buckyballs and the egg. These models are made exclusively from triangle gyroscope modules, so you can make the module any size you like and two of them will still lock together. We like to divide an 8.5" x 11" sheet into strips after first dividing the 8.5" edge into halves or fourths, giving triangles with altitudes of 4.25" or 2.125" (p. 52). These are convenient sizes for modules, are not too hard to fold, and produce manageably sized models. It is also convenient to divide the 11" edge into eighths. These modules are harder to fold and the model is more difficult to assemble. You may also start with squares, or make a triangle tessellation (p. 51).

For each of the gyroscoped Archimedeans, gyroscoped buckyballs, gyroscoped hypothetical buckyballs and the gyroscoped egg you will need two or three different modules. The modules for a particular model cannot, in general, all be constructed from squares of the same size; if they are, the modules will not fit together. There are, however, some cases in which you may start with squares of the same size, e.g. if the model is a buckyball of pentagons and hexagons only.

We now discuss our methods for constructing modules that fit together.

We have come up with three ready-to-use module sizing sets. If you want to combine different modules in the same model, use sizes from the same sizing set. More generally, we also give basic sizing relationships.

Sizing set 1: This sizing set was chosen because 3" squares and strips are convenient to use.

1. triangle module from 3" strip (altitude 3")

2. square module from 3" square

3. pentagon module from 4.25" square (diagonal of pentagon 4.25")

4. hexagon module from 4.25" square (diagonal of hexagon 4.25")

5. octagon module from 10.2" square (starting square 10.2")

6. decagon module from 15.25" square (distance between opposite edges 12.8")


Sizing set 2: This sizing set was chosen because 8.5" squares are the largest that can conveniently be made from 8.5" x 11" paper. The largest module is made from one of these squares.

1. triangle module from 1.6875" (1 11/16") strip (altitude 1.6875")

2. square module from 1.6875" (1 11/16") square

3. pentagon module from 2.4" square (diagonal of pentagon 2.4")

4. hexagon module from 2.4" square (diagonal of hexagon 2.4")

5. octagon module from 5.75" square (starting square 5.75")

6. decagon module from 8.5" square (distance between opposite edges 7.14")


Sizing set 3: This sizing set was chosen because 2.125" (2 1/8") is one fourth of 8.5" and this is convenient for making strips.

1. triangle module from 2.125" strips (8.5" x 11" sheet divided into four strips)

2. square module from 2.125" square (8.5" divided by 4)

3. pentagon module from 3" square (diagonal of pentagon 3")

4. hexagon module from 3" square (diagonal of hexagon 3")

5. octagon module from 7.25" square (starting square 7.25")

6. decagon module from 10.8" square (distance between opposite edges 9.1")


(Continues...)

Excerpted from Multimodular Origami Polyhedra by Rona Gurkewitz, Bennett Arnstein, Bill Quinnell. Copyright © 2003 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Contents

Title Page,
Copyright Page,
INTRODUCTION,
I. BACKGROUND MATERIAL,
A. ARCHIMEDEAN SOLIDS,
B. DUALITY,
C. GYROSCOPED Archimedeans,
D. GYROSCOPE MODULES,
E. BUCKYBALLS,
F. THE EGGS,
II. GALLERY OF POLYHEDRA,
A. THE EGG AND THE GYROSCOPED EGG,
B. ARCHIMEDEANS, CATALANS, AND - GYROSCOPED ARCHIMEDEANS,
C. SEEDS FOR GROWING BUCKYBALLS, BUCKYBALLS, - HYPOTHETICAL BUCKYBALLS, AND THEIR GYROSCOPED FORMS,
III. MODEL CONSTRUCTION,
A. MODEL SELECTION,
B. MODULE SELECTION,
C. MODULE SIZING,
D. NETS AS GUIDES; NAMING SYSTEM FOR THE POLYHEDRA,
E. MODEL ASSEMBLY,
F. FOUR-SIDED DISPLAY STAND FROM A PENTAGON WATERBOMB BASE,
IV POLYGON CONSTRUCTION,
A. SYMBOLS,
B. HOW TO MAKE AN EQUILATERAL TRIANGLE,
C. HOW TO MAKE A PENTAGON,
D. HOW TO MAKE A HEXAGON,
E. HOW TO MAKE AN OCTAGON,
F. HOW TO MAKE A DECAGON,
V MODULE CONSTRUCTION FROM POLYGONS,
A. BASIC TAB AND POCKET LOCK,
B. LARGE MODELS: HEXAGON/PENTAGON POCKET AND CONNECTORS,
VI. REFERENCES AND ACKNOWLEDGMENTS,
Books and articles,
Websites,
Author Contact Information,

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  • Anonymous

    Posted December 21, 2013

    Excellent Intermediate Modular Work.

    With the caveat that careful measurement along with fairly precise cutting will be required for construction of the models, this is an enormously fun and surprisingly informative book.

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  • Anonymous

    Posted April 1, 2013

    Absolutely horrible

    Doesnt show you how to make anything thi is relly stupid

    0 out of 2 people found this review helpful.

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