A Panoply of Polygons presents and organizes hundreds of beautiful, surprising and intriguing results about polygons with more than four sides. (A Cornucopia of Quadrilaterals, a previous volume by the same authors, thoroughly explored the properties of four-sided polygons.) This panoply consists of eight chapters, one dedicated to polygonal basics, the next ones dedicated to pentagons, hexagons, heptagons, octagons and many-sided polygons. Then miscellaneous classes of polygons are explored (e.g., lattice, rectilinear, zonogons, cyclic, tangential) and the final chapter presents polygonal numbers (figurate numbers based on polygons). Applications, real-life examples, and uses in art and architecture complement the presentation where many proofs with a visual nature are included. A Panoply of Polygons can be used as a supplement to a high school or college geometry course. It can also be used as a source for group projects or extra-credit assignments. It will appeal, and be accessible to, anyone with an interest in plane geometry. Claudi Alsina and Roger Nelsen are, jointly and individually, the authors of thirteen previous MAA/AMS books. Those books, and this one, celebrate and illuminate the power of visualization in learning, teaching, and creating mathematics.
Author(s): Claudi Alsina, Roger B. Nelsen
Series: AMS/MAA | Dolciani Mathematical Expositions, 58
Edition: 1
Publisher: American Mathematical Society
Year: 2023
Language: English
Commentary: 2020 Mathematics Subject Classification. Primary 51M05, 51M15, 97G40.
Pages: 267
City: Providence, Rhode Island
Tags: Polygons; Geometry; Real Geometry; Complex Geometry; Euclidean Geometries; Geometric Constructions; Plane Geometry; Solid Geometry
Contents
Preface ix
1 Polygon Basics 1
1.1 Introduction 1
1.2 Polygon preliminaries 2
1.3 Diagonals of convex polygons 5
1.4 Regular polygons 7
1.5 Drawing regular polygons and the Gauss-Wantzel
theorem 14
1.6 Star polygons 16
1.7 Polygonal tiling 19
1.8 Voronoi diagrams and dual tilings 23
1.9 Challenges 25
2 Pentagons 29
2.1 Introduction 29
2.2 Regular pentagons 30
2.3 Drawing a regular pentagon 35
2.4 Folding a regular pentagon 38
2.5 Six types of general convex pentagons 39
2.6 Pentagonal tilings 43
2.7 Pentagrams 48
2.8 Pentagons in space 52
2.9 Miscellaneous examples 54
2.10 Pentagons in architecture 56
2.11 Challenges 57
3 Hexagons 61
3.1 Introduction 61
3.2 Regular hexagons 63
3.3 Cyclic hexagons 71
3.4 Hexagonal tilings 75
3.5 Parahexagons 77
3.6 The carpenter’s square 81
3.7 L-polyominoes 82
3.8 Hexagrams 86
3.9 Miscellaneous examples 90
3.10 Hexagons in architecture 95
3.11 Challenges 97
4 Heptagons 103
4.1 Introduction 103
4.2 Regular heptagons 105
4.3 The diagonals of a regular heptagon 105
4.4 The heptagonal triangle 107
4.5 Drawing a regular heptagon 109
4.6 A neusis construction 112
4.7 Heptagonal tilings 114
4.8 Star heptagons 115
4.9 Heptagons in architecture 116
4.10 Challenges 118
5 Octagons 121
5.1 Introduction 121
5.2 Regular octagons 122
5.3 General convex octagons 128
5.4 Star octagons 131
5.5 Octagons in space 133
5.6 Octagons in architecture 134
5.7 Challenges 137
6 Many-sided Polygons 141
6.1 Introduction 141
6.2 Nonagons 143
6.3 Decagons 148
6.4 Hendecagons 153
6.5 Dodecagons 155
6.6 Gauss and heptadecagons 161
6.7 Archimedes and 24-, 48-, and 96-gons 162
6.8 The 257-gons and the 65537-gons 166
6.9 Miscellaneous many-sided polygons 167
6.10 Challenges 170
7 Miscellaneous Classes of Polygons 175
7.1 Introduction 175
7.2 Lattice polygons 175
7.3 Rectilinear polygons 181
7.4 Zonogons 183
7.5 Cyclic polygons 186
7.6 Tangential polygons 190
7.7 Bicentric polygons 194
7.8 Challenges 195
8 Polygonal Numbers 197
8.1 Introduction 197
8.2 Ordinary polygonal numbers 200
8.3 Centered polygonal numbers 205
8.4 Other figurate numbers derived from polygons 207
8.5 Challenges 209
Solutions to the Challenges 213
Chapter 1 213
Chapter 2 215
Chapter 3 219
Chapter 4 226
Chapter 5 230
Chapter 6 234
Chapter 7 240
Chapter 8 242
Credits and Permissions 247
Bibliography 253
Index 261
