A Panoply of Polygons

281 Pages·2023·47.664 MB·English

by  Claudi Alsina, Roger B. Nelsen

I
AMS/ MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 58
A Panoply of Polygons

I
AMS/ MAA DOLCIANI MATHEMATICAL EXPOSITIONS
VOL58
A Panoply of Polygons
Claudi Alsina
Roger B. Nelsen
MAA PRESS
~ ~~~pr:t:~~-;:;ERICAN
,..,.. ~~: MATHEMATICAL
...
•:~ SOCIElY
Providence, Rhode Island
Dolciani Mathematical Expositions Editorial Board
C. Ray Rosentrater, Editor
Priscilla S. Bremser Heather A. Dye Kim Regnier Jongerius
Alfred M. Dahma C. L. Frenzen Katharine Ott
Matt Delong Jennifer R. Galovich Tom Richmond
2020 Mathematics Subject Classification. Primary 51M05, 51M15, 97G40.
For additional information and updates on this book, visit
www .ams.org/bookpages/d ol-58
Library of Congress Cataloging-in-Publication Data
Names: Alsina, Claudi, author. I Nelsen, Roger B., author.
Title: A panoply of polygons / Claudi Alsina, Roger B. Nelsen.
Description: Providence, Rhode Island : MAA Press, an imprint of the American
Mathematical Society, [2023] I Series: Dolciani mathematical expositions
volume 58 I Includes bibliographical references and index.
Identifiers: LCCN 2022046870 I ISBN 9781470471842 (paperback)
ISBN 9781470472948 (pdf)
Subjects: LCSH: Polygons. I AMS: Geometry - Real and complex geometry -
Euclidean geometries (general) and generalizations. I Geometry - Real and
complex geometry - Geometric constructions. I Mathematics education - Ge
ometry - Plane and solid geometry.
Classification: LCC QA482 .A476 2023 I DDC 516/.154-dc23/eng20230112
LC record available at https://Iccn.loc.gov/2022046870
Copying and reprinting. Individual readers of this publication, and nonprofit li
braries acting for them, are permitted to make fair use of the material, such as to
copy select pages for use in teaching or research. Permission is granted to quote brief
passages from this publication in reviews, provided the customary acknowledgment of
the source is given.
Republication, systematic copying, or multiple reproduction of any material in this
publication is permitted only under license from the American Mathematical Society.
Requests for permission to reuse portions of AMS publication content are handled
by the Copyright Clearance Center. For more information, please visit www.ams.org/
pub 1 ic ati ons /pubpe rmi ssions.
Send requests for translation rights and licensed reprints to reprint-permission©
ams .org.
© 2023 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
§ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at https://www.ams.org/
10 9 8 7 6 5 4 3 2 1 28 27 26 25 24 23
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 ____________________________________________________________________1 9
1.8 Voronoi diagrams and dual tilings ___________________________________ 2 3
1.9 Challenges ___________________________________________________________________________ 2 5
2 Pentagons ______________________________________________________________________________________ 2 9
2. 1 Introduction_ ________________________________________________________________________ 2 9
2. 2 Regular pentagons ______________________________________________________________3 O
2. 3 Drawing a regular pentagon ______________________________________________3 5
2.4 Folding a regular pentagon _______________________________________________ 3 8
2.5 Six types of general convex pentagons ___________________________3 9
2. 6 Pentagonal tilings _______________________________________________________________4 3
2. 7 Pentagrams __________________________________________________________________________4 8
2.8 Pentagons in space ____________________________________________________________ 5 2
2. 9 Miscellaneous examples ___________________________________________________ 5 4
2. 1 O Pentagons in architecture _________________________________________________ 5 6
2. 11 Challenges ___________________________________________________________________________ 5 7
V
VI CONTENTS
3 Hexagons _______________________________________________________________________________________ 61
3 .1 Introduction ------------------------------------------------------------------------61
3. 2 Regular hexagons ________________________________________________________________ 6 3
3 .3 Cyclic hexagons ___________________________________________________________________ 71
3. 4 Hexagonal tilings_ ________________________________________________________________ 7 5
3. 5 Parah exagon s ___________________________________________________________________ __? 7
3 .6 The carpenter's square ____________________________________________________ 81
3. 7 L-po ly omino es _____________________________________________________________________8 2
3 .8 Hexagrams ___________________________________________________________________________ 8 6
3. 9 Miscellaneous examples ___________________________________________________ 9 0
3 .1 O Hexagons in architecture ___________________________________________________ 9 5
3 .11 Challenges ___________________________________________________________________________ 9 7
4 Heptagons __________________________________________________________________________________ 103
4 .1 Introduction ______________________________________________________________________ 1 O3
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-------------------------------------------10 9
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_ ______________________________________________________________________________________1 21
5 .1 Introduction ______________________________________________________________________ 121
5. 2 Regular octagons_ ______________________________________________________________ 12 2
5 .3 General convex octagons_ ________________________________________________ 12 8
5 .4 Star octagons_ ____________________________________________________________________1 31
5 .5 Octagons in space _____________________________________________________________ 13 3
5 .6 Octagons in architecture_ _________________________________________________ 13 4
CONTENTS vu
5. 7 Challenges_ _________________________________________________________________________ 13 7
6 Many-sided Polygons ________________________________________________________________ 141
6.1 Introduction 141
6. 2 No nagons ___________________________________________________________________________ 143
6.3 Decagons ____________________________________________________________________________ 148
6. 4 Hen decagons _____________________________________________________________________ 15 3
6.5 Do decagons _______________________________________________________________________ 15 5
6.6 Gauss and heptadecagons ______________________________________________ 161
6.7 Archimedes and 24-, 48-, and 96-gons _________________________ 162
6.8 The 25 7-gons and the 65 53 7-gons _______________________________1 66
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 Zonogon s ___________________________________________________________________________ 183
7 .5 Cyclic polygons _________________________________________________________________ 186
7 .6 Tangential polygons _________________________________________________________ 190
7. 7 Bi centric polygons_ ___________________________________________________________ 194
7 .8 Challenges _________________________________________________________________________ 195
8 Polygonal Numbers ___________________________________________________________________ 197
8.1 Introduction 197
8. 2 Ordinary polygonal numbers_ _________________________________________ 20 0
8.3 Centered polygonal numbers _________________________________________2 0 5
8.4 Other figurate numbers derived from polygons_ _________ 207
8. 5 Challenges_ _________________________________________________________________________ 20 9
Vlll CONTENTS
Solutions to the Challenges ------------------------------------------------------------·213
Chapter 1 ------------------------------------------------------------------------------------213
Chapter 2 _____________________________________________________________________________________2 15
Chapter 3 ____________________________________________________________________________________ 219
Chapter 4 _____________________________________________________________________________________ 2 2 6
Chapter 5 ------------------------------------------------------------------------------------23 O
Chapter 6 _____________________________________________________________________________________ 23 4
Chapter 7 -------------------------------------------------------------------------------------24 0
Chapter 8 _____________________________________________________________________________________ 24 2
Credits and Permissions 247
Bibliography -------------------------------------------------------------------------------------25 3
Index 261
Preface
Geometry is the archetype oft he beauty of the world.
Johannes Kepler
Geometry is nothing at all, if not a branch ofa rt.
Julian Lowell Coolidge
Beauty is geometry.
J. K. Rowling
To be a well-educated person, the study of geometry should be
included in one's academic background. And some of the most
enchanting objects in elementary geometry are the polygons, figures
constructed from two basic geometric concepts, line segments and
angles. They are prime examples of great beauty created from the
simplest of mathematical objects, and are the subject of this Panoply.
What is a Panoply? Dictionary definitions include a splendid or
striking array or arrangement and a complete or impressive
collection or display of something. The objects in our Panoply are,
for the most part, planar polygons with five or more sides. The
restriction to five or more sides is because entire books can be and
have been written about triangles (e.g., several books in Euclid's
Elements) and quadrilaterals (e.g., the 2020 AMS-MAA book A
Cornucopia of Quadrilaterals). But books devoted to polygons with
five or more sides seem to be rather rare.
Polygons come in a variety of shapes and possess a multitude of
different properties, as witnessed by the many adjectives used to de
scribe them: simple, complex, concave, convex, compound, equian
gular, equilateral, regular, skew, star, cyclic, tangential, bicentric,
rectilinear, lattice, equable, etc. In addition we have the adjective