This easy-to-read book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, the mathematics of polyhedra, combinatorial geometry, and group theory. Using a systematic paper-folding procedure it is possible to construct a regular polygon with any number of sides. This remarkable algorithm has led to interesting proofs of certain results in number theory, has been used to answer combinatorial questions involving partitions of space, and has enabled the authors to obtain the formula for the volume of a regular tetrahedron in around three steps, using nothing more complicated than basic arithmetic and the most elementary plane geometry. All of these ideas, and more, reveal the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions, including clear illustrations, enable the reader to gain hands-on experience constructing these models and to discover for themselves the patterns and relationships they unearth.
Author(s): Peter Hilton, Jean Pedersen, Sylvie Donmoyer
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 309
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 13
Acknowledgments......Page 17
1.1 Four scientists at play......Page 19
1.2 What are flexagons?......Page 21
Basic instructions......Page 22
1.4 Octaflexagons......Page 29
Material error......Page 35
Geometrical error......Page 36
2.2 Some ancient threads......Page 38
Why do we get equilateral triangles?......Page 39
2.4 Does this idea generalize?......Page 42
Is there a general pattern to all this?......Page 51
2.5 Some bonuses......Page 55
Uses for our folded tape......Page 56
3.2 Why does the FAT algorithm work?......Page 57
3.3 Constructing a 7-gon......Page 61
3.4 Some general proofs of convergence......Page 65
4.1 Folding numbers......Page 70
4.2 Recognizing rational numbers of the form ta - 1/tb - 1......Page 76
Case 2......Page 80
4.3 Numerical examples......Page 81
Tidbit 1......Page 84
Tidbit 2......Page 86
5.1 An intuitive approach to polyhedra......Page 89
5.2 Constructing polyhedra from nets......Page 90
General instructions......Page 91
General instructions (alternative)......Page 93
Alternative construction......Page 96
Practical hints......Page 97
5.3 What is a regular polyhedron?......Page 98
6.1 Preparing the pattern piece for a pentagonal dipyramid......Page 104
6.2 Assembling the pentagonal dipyramid......Page 105
6.3 Refinements for dipyramids......Page 106
6.4 Constructing braided rotating rings of tetrahedra......Page 108
Why do you suppose it was possible to make this model?......Page 111
6.6 More fun with rotating rings......Page 112
Some history about rotating rings......Page 113
7.1 Constructing an 11-gon......Page 114
7.2 The quasi-order theorem......Page 118
7.3 The quasi-order theorem when t = 3......Page 122
7.4 Paper-folding connections with various famous number sequences......Page 123
7.5 Finding the complementary factor and reconstructing the symbol......Page 124
8.1 Facts of life......Page 128
8.2 Description of the puzzle......Page 129
8.3 How to make the puzzle pieces......Page 130
8.4 Assembling the braided tetrahedron......Page 133
8.5 Assembling the braided octahedron......Page 134
8.6 Assembling the braided cube......Page 135
Volumes of some related polyhedra......Page 136
9.1 A curious fact......Page 141
9.2 Preparing the strips......Page 144
9.4 Braiding the golden dodecahedron......Page 147
9.5 Braiding the dodecahedron......Page 149
9.6 Braiding the icosahedron......Page 152
9.7 Constructing more symmetric tetrahedra, octahedra, and icosahedra......Page 155
9.8 Weaving straight strips on other polyhedral surfaces......Page 157
10.1 Symmetries of the cube......Page 163
10.2 Symmetries of the regular octahedron and regular tetrahedron......Page 167
The diagonal cube has some combinatorial surprises......Page 169
Remark on orientation and symmetry......Page 170
10.3 Euler's formula and Descartes' angular deficiency......Page 172
10.4 Some combinatorial properties of polyhedra......Page 176
11.1 How can there be more dodecahedra?......Page 181
Instructions......Page 183
Instructions......Page 186
Instructions......Page 189
11.5 Magical relationships between special dodecahedra......Page 191
12.1 What is a collapsoid?......Page 193
12.2 Preparing the cells, tabs, and flaps......Page 194
Cells......Page 195
Flaps......Page 196
12.3 Constructing a 12-celled polar collapsoid......Page 197
Practical hints......Page 199
12.4 Constructing a 20-celled polar collapsoid......Page 200
12.5 Constructing a 30-celled polar collapsoid......Page 201
12.6 Constructing a 12-celled equatorial collapsoid......Page 202
For the 12-celled collapsoid (polar or equatorial)......Page 203
12.8 How do we find other collapsoids?......Page 204
13.2 How to build the special trihexaflexagon......Page 213
13.3 The happy group......Page 215
13.4 The entire group......Page 218
13.6 What next?......Page 221
14.2 Divisions of the plane......Page 224
Case 1......Page 225
Case 2......Page 226
14.3 Some facts about the Platonic solids......Page 228
Tetrahedron......Page 230
Octahedron......Page 231
Dodecahedron......Page 236
Icosahedron......Page 238
14.5 More general questions......Page 240
15.1 Polya's speculation......Page 241
15.2 Polya's dream......Page 242
15.3 …The dream comes true......Page 247
15.4 Further generalizations......Page 250
16.1 Symmetry: A really big idea......Page 254
Definition of a group......Page 257
Remarks......Page 261
16.3 Homologues......Page 265
16.4 The Polya enumeration theorem......Page 266
Definition of even and odd permutations......Page 271
Step 3......Page 272
16.6 Epilogue: Polya and ourselves – Mathematics, tea, and cakes......Page 274
17.2 The coach theorem......Page 278
17.3 The generalized quasi-order theorem......Page 282
17.4 The generalized coach theorem......Page 285
17.5 Parlor tricks......Page 289
17.6 A little linear algebra......Page 293
17.7 Some open questions......Page 299
References......Page 300
Index......Page 304
