Flexagons Inside Out

Flexagons, paper models that can be bent in different ways to change their shape, are easy to make and work in surprising ways. This book contains numerous diagrams that the reader can photocopy and use to construct a variety of fascinating flexagons. The author also explains the mathematics behind these amazing creations. Although knowledge of the technical details requires a mathematical background, the models can be made and used by anyone. Flexagons appeals to all readers interested in puzzles and recreational mathematics.

Preface

ix

List of notation

xi

Introduction

1

(4)

1 Malting and flexing flexagons

5

(9)

1.1 The trihexaflexagon

5

(1)

1.2 The pinch flex

6

(1)

1.3 A simple square flexagon

7

(1)

1.4 General assembly instructions

8

(5)

1.5 Decoration of faces

13

(1)

2 Early history of flexagons

14

(4)

2.1 The discovery of flexagons

14

(1)

2.2 The Flexagon Committee

15

(1)

2.3 Revival of interest

16

(2)

3 Geometry of flexagons

18

(13)

3.1 Flexagon characteristics

19

(1)

3.2 Appearance of main positions

20

(1)

3.3 Flexagons as linkages

21

(3)

3.4 Mapping flexagons

24

(5)

3.5 Flexagon figures and symbols

29

(2)

4 Hexaflexagons

31

(16)

4.1 Degrees of freedom of hexaflexagons

32

(1)

4.2 The trihexaflexagon revisited

33

(3)

4.3 Multicycle hexaflexagons

36

(4)

4.4 Hexaflexagon design

40

(2)

4.5 Hexaflexagon face arrangements

42

(5)

5 Hexaflexagon variations

47

(6)

5.1 Triangle flexagons

47

(2)

5.2 The Vflex

49

(3)

5.3 Origami with hexaflexagons

52

(1)

6 Square flexagons

53

(29)

6.1 Single cycle square flexagons

54

(4)

6.2 Reconstruction from a flexagon figure

58

(1)

6.3 Square flexagons with main position links

59

(10)

6.4 Distinct types of square flexagon with main position links

69

(1)

6.5 Square flexagons with box position links

70

(6)

6.6 Square flexagon face arrangements

76

(2)

6.7 The interleaf flex

78

(4)

7 Introduction to convex polygon flexagons

82

(14)

7.1 The convex polygon family

83

(1)

7.2 Degrees of freedom of intermediate positions

84

(3)

7.3 The digon flexagon

87

(3)

7.4 Triangle flexagons revisited

90

(3)

7.5 Square flexagons revisited

93

(3)

8 Typical convex polygon flexagons

96

(23)

8.1 Characteristics of typical convex polygon flexagons

97

(1)

8.2 Pentagon flexagons

98

(8)

8.3 Hexagon flexagons

106

(4)

8.4 Octagon flexagons

110

(9)

9 Ring flexagons

119

(18)

9.1 Star flexagons

120

(1)

9.2 The ring flexagon family

121

(2)

9.3 Compound flexagons

123

(3)

9.4 Compound square flexagons

126

(2)

9.5 Compound pentagon flexagons

128

(4)

9.6 Compound hexagon flexagons

132

(5)

10 Distorted polygon flexagons

137

(17)

10.1 Distorted polygons

137

(3)

10.2 Some distorted polygon flexagons

140

(9)

10.3 Right angle triangle flexagons

149

(5)

11 Flexahedra

154

(10)

11.1 Four dimensions

154

(1)

11.2 Prism flexahedra

155

(2)

11.3 Pyramid flexahedra

157

(1)

11.4 Cube flexahedra

158

(6)

References

164

(3)

Flexagon index

167

(2)

Subject index

169

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

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Summary

Table of Contents

Supplemental Materials