Flexagons Inside Out

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Notation......Page 13
Introduction......Page 15
1.1 The trihexaflexagon......Page 19
1.2 The pinch flex......Page 20
1.3 A simple square flexagon......Page 21
1.4 General assembly instructions......Page 22
1.5 Decoration of faces......Page 27
2.1 The discovery of flexagons......Page 28
2.2 The Flexagon Committee......Page 29
2.3 Revival of interest......Page 30
3 Geometry of flexagons......Page 32
3.1 Flexagon characteristics......Page 33
3.2 Appearance of main positions......Page 34
3.3 Flexagons as linkages......Page 35
3.3.1 Sarrut’s parallel motion......Page 36
3.4.1 Full map of the irregular single cycle square flexagon......Page 38
3.4.3 Pat structure of the irregular single cycle square flexagon......Page 41
3.4.4 Some other types of map......Page 42
3.5 Flexagon figures and symbols......Page 43
4 Hexaflexagons......Page 45
4.1.1 The null hexaflexagon......Page 46
4.2 The trihexaflexagon revisited......Page 47
4.3.2 The four faced hexaflexagon......Page 50
4.3.3 The Tuckerman traverse......Page 52
4.3.4 The hexahexaflexagon......Page 53
4.4 Hexaflexagon design......Page 54
4.4.2 The synthetic method......Page 55
4.5.1 A seven faced hexaflexagon......Page 56
4.5.2 A puzzle hexahexaflexagon......Page 58
5.1 Triangle flexagons......Page 61
5.1.1 The single cycle triangle flexagon......Page 62
5.2.1 V-flexing the hexahexaflexagon......Page 63
5.3 Origami with hexaflexagons......Page 66
6 Square flexagons......Page 67
6.1.2 The regular single cycle square flexagon......Page 68
6.1.3 The incomplete single cycle square flexagon revisited......Page 69
6.2 Reconstruction from a flexagon figure......Page 72
6.3.1 Main position links......Page 73
6.3.2 The square flexagon with two regular cycles and a main position link......Page 74
6.3.3 Square flexagons with regular cycles and main position links......Page 76
6.3.5 The square flexagon with one regular cycle and one irregular cycle and a main position link......Page 77
6.3.6 Deleting faces from square flexagons......Page 78
6.3.7 The square flexagon with one regular cycle and one incomplete cycle and a main position link......Page 79
6.3.9 Square flexagons with two incomplete cycles and a main position link......Page 80
6.4 Distinct types of square flexagon with main position links......Page 83
6.5 Square flexagons with box position links......Page 84
6.5.2 Square flexagons with two incomplete cycles and a pair of box position links......Page 85
6.5.3 The square flexagon with two irregular cycles and a pair of box position links......Page 86
6.5.4 A puzzle square flexagon......Page 88
6.6.1 A five faced square flexagon......Page 90
6.7 The interleaf flex......Page 92
6.7.1 Interleaf flexing a square flexagon with two incomplete cycles and a main position link......Page 93
7 Introduction to convex polygon flexagons......Page 96
7.1 The convex polygon family......Page 97
7.2 Degrees of freedom of intermediate positions......Page 98
7.3.1 The truncated digon flexagon......Page 101
7.3.2 The double hinged truncated digon flexagon......Page 102
7.3.3 The flexatube......Page 103
7.4 Triangle flexagons revisited......Page 104
7.4.1 The single cycle truncated triangle flexagon......Page 105
7.5.1 The regular single cycle truncated square flexagon......Page 107
8 Typical convex polygon flexagons......Page 110
8.1.1 Single complete principal cycle convex polygon flexagons......Page 111
8.2.1 Types of single principal cycle pentagon flexagon......Page 112
8.2.2 The regular single principal cycle pentagon flexagon (5, 5)......Page 113
8.2.5 Pentagon flexagon with two incomplete principal cycles and a pair of subsidiary main position links......Page 116
8.3.1 Types of single principal cycle hexagon flexagon......Page 120
8.3.2 The regular single principal cycle hexagon flexagon......Page 122
8.3.3 An irregular single principal cycle hexagon flexagon......Page 123
8.4 Octagon flexagons......Page 124
8.4.1 The regular single principal cycle octagon flexagon (8, 8)......Page 127
8.4.2 The three faced octagon flexagon......Page 129
8.4.3 The twist flex......Page 132
9 Ring flexagons......Page 133
9.1 Star flexagons......Page 134
9.2 The ring flexagon family......Page 135
9.3 Compound flexagons......Page 137
9.3.1 Modified flexagon figures for compound flexagons......Page 139
9.4 Compound square flexagons......Page 140
9.4.1 The 3-cycle compound square flexagon......Page 141
9.5 Compound pentagon flexagons......Page 142
9.5.1 The 3-cycle compound pentagon flexagon......Page 145
9.6 Compound hexagon flexagons......Page 146
9.6.1 The principal 4-cycle compound hexagon flexagon......Page 150
10.1 Distorted polygons......Page 151
10.2.2 A four faced rhombus flexagon with three sectors......Page 154
10.2.3 A four faced rhombus flexagon with two sectors......Page 157
10.2.4 The 3-cycle compound rhombus flexagon......Page 158
10.2.5 A six faced distorted hexagon flexagon......Page 160
10.3 Right angle triangle flexagons......Page 163
10.3.1 The three faced right angle triangle flexagon......Page 165
10.3.2 A four faced right angle triangle flexagon......Page 167
11.1 Four dimensions......Page 168
11.2 Prism flexahedra......Page 169
11.2.1 A three hyperfaced prism flexahedron......Page 170
11.3 Pyramid flexahedra......Page 171
11.4 Cube flexahedra......Page 172
11.4.1 A four hyperfaced cube flexahedron......Page 173
11.4.2 A four hyperfaced truncated cube flexahedron......Page 174
11.4.3 A three hyperfaced cube flexahedron......Page 175
11.4.4 A five hyperfaced cube flexahedron......Page 176
References......Page 178
Flexagon index......Page 181
Subject index......Page 183