Pentagons and Pentagrams: An Illustrated History

201 Pages·2022·29.731 MB·English

by  Eli Maor, Eugen Jost

PENTAGONS AND PENTAGRAMS

PENTAGONS

and
PENTAGR AMS

An Illustrated History

TEXT BY ILLUSTRATIONS BY
ELI MAOR EUGEN JOST
PRINCE TON UNIVERSITY PRESS
Prince ton & Oxford
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Library of Congress Cataloging-in-Publication Data
Names: Maor, Eli, author. | Jost, Eugen, 1950– illustrator.
Title: Pentagons and pentagrams : an illustrated history /
text by Eli Maor ; illustrations by Eugen Jost.
Description: Princeton : Princeton University Press, [2022] |
Includes bibliographical references and index.
Identifiers: LCCN 2021041790 | ISBN 9780691201122 (hardback)
Subjects: LCSH: Pentagon. | Pentacles. | Mathematics—History. |
BISAC: MATHEMATICS / Geometry / General | MATHEMATICS /
History & Philosophy
Classification: LCC QA482 .M34 2022 | DDC 516/.154—dc23/eng/20211112
LC record available at https://lccn.loc.gov/2021041790
British Library Cataloging-i n- Publication Data is available
Editorial: Hallie Stebbins, Kristen Hop, and Kiran Pandey
Production Editorial: Natalie Baan
Text and Jacket Design: Chris Ferrante
Production: Danielle Amatucci
Publicity: Kate Farquhar-T homson and Matthew Taylor
Copyeditor: Jennifer McClain
Jacket Credits: Front: Eugen Jost, Untitled. Back: Eugen Jost,
The Golden Ratio.
Frontispiece: Leonardo da Vinci, Dodecahedron
This book has been composed in New Century Schoolbook LT and Futura PT
Printed on acid- free paper. ∞
Printed in the United States of Amer i ca
10 9 8 7 6 5 4 3 2 1
To my dear wife, Dalia, whose love and support
has meant so much to me over all these years
—E LI MAOR
To my grandchildren— Julian, Valentin, Laurin,
Carla, Elina, and Rafael— the joy of my old days
—E UGEN JOST

CONTENTS
PREFACE. Why the Pentagon? ix
1 Five 1
2 ϕ 14
SIDEBAR 1. Four Famous Irrational Numbers:
2,, ϕ , e, and π 21
3 But Is It Divine? 26
4 Constructing the Regular Pentagon 38
SIDEBAR 2. Pentagonal Numbers 55
5 The Pentagram 58
CENTERPIECE. Color Plates
6 Pentastars 71
SIDEBAR 3. Pentagonal Puzzles and Curiosa 78
7 Tessellations: Defying the Impossible 88
8 The Discovery of Fivefold Symmetry in Crystals 103
SIDEBAR 4. An Unsolved Mystery 115
9 Oh, That Pentagon 119
APPENDIXES
A The Elementary Euclidean Constructions 133
B Three Properties of the Fibonacci Numbers 135
C A Proof That There Exist Only Five Platonic Solids 141
D Summary of Formulas 147
E Solutions to Puzzles 151
BIBLIOGRAPHY 157
CREDITS 159
INDEX 163

PREFACE
Why the Pentagon?
Why the pentagon—no, not the defence
headquarters, but its eponym, the five- sided
polygon— continues to pester mathematicians.
— RICHARD MURCHIE1
ON SEPTEMBER 14, 1959, the Soviet spacecraft Luna 2 crash-
landed on the Moon east of Mare Imbrium and became the
first human- made artifact to reach another world. Upon
impact, the spacecraft scattered around a cluster of small,
pentagonal- shaped metal pennants with the hammer and
sickle, the insignia of the Soviet Union (figure 0.1). The
pentagon—t his ancient sign of brotherhood— thus became
humanity’s greeting card on its first venture into the solar
system.
FIGURE 0.1. Luna 2