Serious Fun with Flexagons (eBook)

A Compendium and Guide

L.P. Pook (Autor)

eBook Download: PDF

2009 | 2009
XVI, 330 Seiten
Springer Netherland (Verlag)
978-90-481-2503-6 (ISBN)

Infos

A flexagon is a motion structure that has the appearance of a ring of hinged polygons. It can be flexed to display different pairs of faces, usually in cyclic order. Flexagons can be appreciated as toys or puzzles, as a recreational mathematics topic, and as the subject of serious mathematical study. Workable paper models of flexagons are easy to make and entertaining to manipulate. The mathematics of flexagons is complex, and how a flexagon works is not immediately obvious on examination of a paper model. Recent geometric analysis, included in the book, has improved theoretical understanding of flexagons, especially relationships between different types.

This profusely illustrated book is arranged in a logical order appropriate for a textbook on the geometry of flexagons. It is written so that it can be enjoyed at both the recreational mathematics level, and at the serious mathematics level. The only prerequisite is some knowledge of elementary geometry, including properties of polygons. A feature of the book is a compendium of over 100 nets for making paper models of some of the more interesting flexagons, chosen to complement the text. These are accurately drawn and reproduced at half full size. Many of the nets have not previously been published. Instructions for assembling and manipulating the flexagons are included.

 

 

Since my formal retirement in 1998 I have been a Visiting Professor in the Department of Mechanical Engineering at University College London, where I do some part time teaching in the fields of metal fatigue and fracture mechanics. I am a senior official of the European Structural Integrity Society, and have been involved with the organisation of two of their conferences (FCP 2003 and CP 2006), including editing conference proceedings and associated special issues of journals.

Inhalt

162654_1_En_FM1_Chapter_O.pdf 2
162654_1_En_1_Chapter_O.pdf 16
Chapter 1 16
Introduction 16
1.1 General Features 16
1.2 Terminology 20
1.3 Outline of Book 24
1.4 Making Flexagons 26
1.4.1 General Assembly Instructions 27
1.4.2 The Two Sector First Order Fundamental Square Even Edge Flexagon 28
References 28
162654_1_En_2_Chapter_O.pdf 30
Chapter 2 30
Polygon Rings 30
2.1 Introduction 30
2.1.1 Multiple Polygons 31
2.1.2 Combinations 32
2.2 Edge Rings of Regular Polygons 32
2.2.1 General Properties 32
2.2.2 Regular Even Edge Rings 35
2.2.3 Regular Odd Edge Rings 36
2.2.4 Compound Edge Rings 38
2.2.5 Irregular Edge Rings 41
2.3 Edge Rings of Irregular Polygons 42
2.3.1 General Properties 42
2.3.2 Even Edge Rings of Silver and Bronze Triangles 43
2.4 Vertex Rings 44
2.4.1 General Properties 44
2.4.2 Vertex Rings of Squares 45
References 46
162654_1_En_3_Chapter_O.pdf 47
Chapter 3 47
Fundamental Nets 47
3.1 Introduction 47
3.2 First Order Fundamental Edge Nets 48
3.3 Second Order Fundamental Edge Nets 48
3.4 Fundamental Vertex Nets 50
3.5 Fundamental Silver and Bronze Edge Nets 55
References 55
162654_1_En_4_Chapter_O.pdf 56
Chapter 4 56
Fundamental Edge Flexagons 56
4.1 Introduction 56
4.1.1 Standard Face Numbering Sequences 57
4.1.2 Truncated Flexagons 57
4.2 First Order Fundamental Even Edge Flexagons 58
4.2.1 General Properties 58
4.2.2 Ring Even Edge Flexagons 63
4.2.3 First Order Fundamental Triangle Even Edge Flexagons 64
4.2.4 First Order Fundamental Square Even Edge Flexagons 66
4.2.5 Detailed Analysis of Flexagons 67
4.2.5.1 Sector Symbols 67
4.2.5.2 Tuckerman Diagrams 68
4.2.5.3 Full Maps 69
4.2.5.4 Flexagon Diagrams 70
4.2.6 First Order Fundamental Pentagon Even Edge Flexagons 71
4.2.7 First Order Fundamental Hexagon Even Edge Flexagons 75
4.2.8 First Order Fundamental Octagon Even Edge Flexagons 77
4.2.9 First Order Fundamental Dodecagon Even Edge Flexagons 79
4.3 Second Order Fundamental Odd Edge Flexagons 81
4.3.1 General Properties 81
4.3.2 Second Order Fundamental Triangle Odd Edge Flexagons 82
4.3.3 A Second Order Fundamental Square Odd Edge Flexagon 85
4.3.4 A Second Order Fundamental 20-gon Odd Edge Flexagon 87
References 88
162654_1_En_5_Chapter_O.pdf 90
Chapter 5 90
Fundamental Skeletal and Point Flexagons 90
5.1 Introduction 90
5.2 First Order Fundamental Even Skeletal Flexagons 91
5.2.1 General Properties 91
5.2.2 First Order Fundamental Triangle Even Skeletal Flexagons 92
5.2.3 A First Order Fundamental Square Even Skeletal Flexagon 94
5.3 Fundamental Point Flexagons 95
5.3.1 General Properties and Unagons 95
5.3.2 The Fundamental Triangle Point Flexagon 96
5.3.3 The Fundamental Square Point Flexagon 97
5.3.4 Fundamental Pentagon Point Flexagons 98
5.3.5 The Fundamental Hexagon Point Flexagon 99
5.4 Interleaved Fundamental Point Flexagons 100
5.4.1 General Properties 100
5.4.2 The Interleaved Fundamental Pentagon Point Flexagon 102
5.4.3 An Interleaved Fundamental Enneagon Point Flexagon 103
5.5 Augmented Fundamental Point Flexagons 104
5.5.1 General Properties 104
5.5.2 An Augmented Fundamental Triangle Point Flexagon 105
5.5.3 An Augmented Fundamental Square Point Flexagon 106
5.6 Augmented Interleaved Fundamental Point Flexagons 108
5.6.1 General Properties 108
5.6.2 Augmented Interleaved Fundamental Triangle Point Flexagons 110
5.6.3 An Augmented Interleaved Fundamental Square Point Flexagon 113
References 114
162654_1_En_6_Chapter_O.pdf 115
Chapter 6 115
Fundamental Compound Edge Flexagons 115
6.1 Introduction 115
6.2 General Properties 116
6.3 Triangular Fundamental Compound Edge Flexagons 119
6.3.1 Some Properties 119
6.3.2 A Fundamental Square Compound Edge Flexagon 119
6.3.3 A Fundamental Pentagon Compound Edge Flexagon 120
6.4 A Square-Like Fundamental Compound Edge Flexagon 121
6.4.1 Some Properties 121
6.4.2 A Fundamental Hexagon Compound Edge Flexagon 121
6.5 Pentagonal Fundamental Compound Edge Flexagons 123
6.5.1 Some Properties 123
6.5.2 A Fundamental Square Compound Edge Flexagon 123
6.5.3 A Fundamental Hexagon Compound Edge Flexagon 124
6.6 A Hexagonal Fundamental Compound Edge Flexagon 125
6.6.1 Some Properties 125
6.6.2 A Fundamental Octagon Compound Edge Flexagon 125
6.7 Heptagonal Fundamental Compound Edge Flexagons 127
6.7.1 Some Properties 127
6.7.2 A Fundamental Hexagon Compound Edge Flexagon 127
6.7.3 A Fundamental Decagon Compound Edge Flexagon 129
References 130
162654_1_En_7_Chapter_O.pdf 131
Chapter 7 131
Irregular Cycle Flexagons 131
7.1 Introduction 131
7.2 Irregular Cycle Even Edge Flexagons 133
7.2.1 General Properties 133
7.2.2 Derivation of Nets 134
7.2.3 The Irregular Cycle Square Even Edge Flexagon 135
7.2.4 An Irregular Cycle Pentagon Even Edge Flexagon 138
7.2.5 Irregular Cycle Hexagon Even Edge Flexagons 139
7.3 Irregular Cycle Interleaved Point Flexagons 141
7.3.1 General Properties 141
7.3.2 Interleaf Flexes 143
7.3.3 The Irregular Cycle Interleaved Square Point Flexagon 143
7.3.4 Irregular Cycle Interleaved Pentagon Point Flexagons 145
7.3.5 Irregular Cycle Interleaved Hexagon Point Flexagons 147
7.3.6 Augmented Irregular Cycle Interleaved Triangle Point Flexagons 149
7.4 Distinct Face Numbering Sequences 153
7.5 Irregular Cycle Non Interleaved Point Flexagons 154
7.5.1 General Properties 154
7.5.2 The Irregular Cycle Non Interleaved Square Point Flexagon 155
7.5.3 An Irregular Cycle Non Interleaved Pentagon Point Flexagon 155
References 157
162654_1_En_8_Chapter_O.pdf 158
Chapter 8 158
Degenerate Flexagons 158
8.1 Introduction 158
8.2 Degenerate Even Edge Flexagons 160
8.2.1 General Properties 160
8.2.2 A Degenerate Square Even Edge Flexagon 162
8.2.3 Degenerate Pentagon Even Edge Flexagons 163
8.2.4 Degenerate Hexagon Even Edge Flexagons 165
8.2.5 Degenerate Octagon Even Edge Flexagons 168
8.2.6 A Degenerate Dodecagon Even Edge Flexagon 170
8.3 Degenerate Non Interleaved Point Flexagons 172
8.3.1 General Properties 172
8.3.2 The Degenerate Non Interleaved Square Point Flexagon 174
8.3.3 Degenerate Non Interleaved Pentagon Point Flexagons 175
8.4 Degenerate Irregular Cycle Interleaved Point Flexagons 176
8.4.1 General Properties 176
8.4.2 Degenerate Interleaved Triangle Point Flexagons 177
8.5 Degenerate Compound Edge Flexagons 180
8.5.1 General Properties 180
8.5.2 The Degenerate Square-Like Hexagon Compound Edge Flexagon 182
8.5.3 Degenerate Pentagonal Compound Edge Flexagons 183
8.5.3.1 A Degenerate Pentagonal Square Compound Edge Flexagon 183
8.5.3.2 A Degenerate Pentagonal Hexagon Compound Edge Flexagon 184
162654_1_En_9_Chapter_O.pdf 186
Chapter 9 186
Irregular Ring Even Edge Flexagons 186
9.1 Introduction 186
9.1.1 Fundamental Irregular Ring Even Edge Flexagons 186
9.2 Irregular Ring Triangle Even Edge Flexagons 188
9.2.1 General Properties 188
9.2.2 An Irregular Ring 12 Triangle Even Edge Flexagon 188
9.2.3 An Irregular Ring Eight Triangle Even Edge Flexagon 189
9.2.4 An Irregular Ring 16 Triangle Even Edge Flexagon 190
9.3 Irregular Ring Square Even Edge Flexagons 190
9.3.1 General Properties 190
9.3.2 Irregular Ring Six Square Even Edge Flexagons 191
9.3.3 Irregular Ring Eight Square Even Edge Flexagons 194
9.4 Irregular Ring Pentagon Even Edge Flexagons 197
9.4.1 General Properties 197
9.4.2 An Irregular Ring Six Pentagon Even Edge Flexagon 198
9.4.3 An Irregular Ring Eight Pentagon Even Edge Flexagon 201
9.5 Irregular Ring Hexagon Even Edge Flexagons 202
9.5.1 General Properties 202
9.5.2 An Irregular Ring Six Hexagon Even Edge Flexagon 203
9.5.3 Irregular Ring Eight Hexagon Even Edge Flexagons 205
9.6 Irregular Ring Dodecagon Even Edge Flexagons 208
9.6.1 General Properties 208
9.6.2 Irregular Ring Eight Dodecagon Even Edge Flexagons 209
162654_1_En_10_Chapter_O.pdf 212
Chapter 10 212
Irregular Polygon Edge Flexagons 212
10.1 Introduction 212
10.1.1 Transformation of Polygons 213
10.1.2 Stretch Flexagons and Stretch Polygon Rings 214
10.2 Irregular Triangle Edge Flexagons 214
10.2.1 Irregular Triangles 214
10.2.2 Isosceles Triangle Edge Rings 215
10.2.3 Bronze Even Edge Rings 217
10.2.4 Isosceles Triangle Even Edge Flexagons 218
10.2.4.1 Fundamental Silver Even Edge Flexagons 218
10.2.4.2 The Irregular Ring Ten Silver Triangle Even Edge Flexagon 221
10.2.4.3 A Partial Overlap Silver Even Edge Flexagon 223
10.2.4.4 A Fundamental Cart Wheel Even Edge Flexagon 224
10.2.4.5 A Fundamental Star Ring Even Edge Flexagon 225
10.2.5 Isosceles Triangle Odd Edge Flexagons 225
10.2.6 Scalene Triangle Even Edge Flexagons 227
10.2.6.1 Fundamental Bronze Even Edge Flexagons 228
10.2.6.2 The Two Sector Fundamental Bronze Even Edge Flexagon 230
10.2.6.3 The Three Sector Fundamental Bronze Even Edge Flexagon 231
10.2.6.4 The Four Sector Fundamental Bronze Even Edge Flexagon 232
10.2.6.5 The Six Sector Fundamental Bronze Even Edge Flexagon 232
10.2.6.6 The Irregular Ring Eight Bronze Triangle Even Edge Flexagon 238
10.2.7 A Partial Overlap Bronze Even Edge Flexagon 239
10.2.8 A Scalene Triangle Even Edge Flexagon 240
10.3 Irregular Quadrilateral Even Edge Flexagons 242
10.3.1 Irregular Quadrilaterals 242
10.3.2 Irregular Quadrilateral Even Edge Rings 243
10.3.3 A Rectangle Even Edge Flexagon 244
10.3.4 Rhombus Even Edge Flexagons 246
10.3.4.1 A 60°–120° Rhombus Even Edge Flexagon 246
10.3.4.2 Irregular Ring Rhombus Even Edge Flexagons 247
10.3.4.3 A Partial Overlap 60°–120° Rhombus Even Edge Flexagon 249
10.3.5 A Trapezium Even Edge Flexagon 250
10.4 Irregular Pentagon Even Edge Flexagons 251
10.4.1 Irregular Pentagons 251
10.4.2 Irregular Pentagon Even Edge Rings 252
10.4.3 An Equiangular Irregular Pentagon Even Edge Flexagon 253
10.4.4 An Irregular Ring Eight Irregular Pentagon Even Edge Flexagon 255
References 255
162654_1_En_11_Chapter_O.pdf 257
11 257
Complex Flexagons 257
11.1 Introduction 257
11.2 Linked Even Edge Flexagons 259
11.2.1 Methods of Linking 259
11.2.2 Linked Hexaflexagons 261
11.2.3 Linked Square Even Edge Flexagons 265
11.2.3.1 Square Even Edge Flexagons with Flat Main Position Links 266
11.2.3.2 Square Even Edge Flexagons with Box Position Links 268
11.2.4 Linked Pentagon Even Edge Flexagons 272
11.2.4.1 A Pentagon Even Edge Flexagon with a Skew Main Position Link 273
11.2.4.2 A Pentagon Even Edge Flexagon with a Pair of Slant Main Position Links 274
11.2.5 Linked Silver Even Edge Flexagons 275
11.2.5.1 General Properties 275
11.2.5.2 A Silver Even Edge Flexagon with a Flat Main Position Link 275
11.2.5.3 An Uncut Silver Even Edge Flexagon with a Flat Main Position Link 277
11.2.5.4 A Silver Even Edge Flexagon with a Skew Main Position Link 279
11.2.6 Linked Bronze Even Edge Flexagons 281
11.2.6.1 General Properties 281
11.2.6.2 A Bronze Even Edge Flexagon with a Skew Main Position Link 281
11.2.6.4 An Uncut Bronze Even Edge Flexagon with a Flat Main Position Link 286
11.4 Conjoined Point Flexagons 292
11.4.1 General Properties 292
11.4.2 Conjoined Triangle Point Flexagons 293
11.4.3 A Conjoined Pentagon Point Flexagon 294
11.5 Bundled Odd Edge Flexagons 294
11.5.1 General Properties 294
11.5.2 A Five Sector Bundled Triangle Odd Edge Flexagon 296
11.5.3 The Seven and Six Flexagon 297
11.6 Slipagons 298
11.6.1 General Properties 298
11.6.2 A Trihexaflexagon Slipagon 298
11.6.3 A Partial Overlap Silver Even Edge Slipagon 300
11.7 Coupled Point Flexagons 302
11.7.1 General Properties 302
11.7.2 A Coupled Triangle Point Flexagon 305
11.7.2.1 An Interleaved Conjoined Triangle Point Flexagon 305
11.7.3 A Coupled Square Point Flexagon 306
References 308
162654_1_En_12_Chapter_O.pdf 309
Chapter 12 309
Miscellaneous Flexagons 309
12.1 Introduction 309
12.2 Three Sector Odd Flexagons 310
12.2.1 General Properties 310
12.2.2 The Three Sector Fundamental Dodecagon Odd Edge Flexagon 310
12.2.3 A Three Sector Isosceles Triangle Odd Edge Flexagon 311
12.2.4 A Three Sector Hexagon Odd Skeletal Flexagon 312
12.3 Degenerate Odd Edge Flexagons 314
12.3.1 General Properties 314
12.3.2 A Degenerate Square Odd Edge Flexagon 314
12.4 Alternating Odd Edge Flexagons 315
12.4.1 General Properties 315
12.4.2 A Square Alternating Odd Edge Flexagon 316
12.5 Flapagons 318
12.5.1 General Properties 318
12.5.2 The Fundamental Square Duplex Flapagon 318
12.5.3 A Square Flapagon–Flexagon Hybrid 319
12.5.4 The Fundamental Isosceles Triangle Triplex Flapagon 319
12.6 Multiplex Edge Flexagons 320
12.6.1 General Properties 320
12.6.2 A Square Duplex Edge Flexagon 321
12.6.3 The Thrice Threefold Flexagon 323
12.6.3.1 Flexing as a Triplex Edge Flexagon 323
12.6.3.2 Transformations and Flexing Using a Threefold Flex 325
12.6.3.3 Transformation and Flexing Using an Asymmetric Fourfold Pinch Flex 328
12.6.3.4 Traversing 3-Cycles by Using Threefold Flexes 330
12.6.3.5 A Trapezoidal Flexagon 330
12.7 A Hooke’s Joint Flexagon 331
References 333
162654_1_En_Index_Chapter_O.pdf 334

Produkt-Details

• Erscheint lt. Verlag: 4.8.2009
• Reihe/Serie: Solid Mechanics and Its Applications
• Reihe/Serie: Solid Mechanics and Its Applications
• Zusatzinfo: XVI, 330 p.
• Verlagsort: Dordrecht
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Calculus • Derivation • flexagons • Geometry • History • History of Mathematics • Mathematics • Model
• ISBN-10: 90-481-2503-0 / 9048125030
• ISBN-13: 978-90-481-2503-6 / 9789048125036