2006 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock,
Oxfordshire OX20 1TW
All Rights Reserved
Fifth printing, and ﬁrst paperback printing, with a new preface by the authors, 2008
Paperback ISBN: 978-0-691-13871-8
The Library of Congress has cataloged the cloth edition of this book as follows
Ash, Avner, 1949–
Fearless symmetry : exposing the hidden patterns of numbers /
Avner Ash and Robert Gross. p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-691-12492-6 (acid-free paper)
ISBN-10: 0-691-12492-2 (acid-free paper)
1. Number theory. I. Gross, Robert, 1959– II. Title.
British Library Cataloging-in-Publication Data is available
About the cover image, courtesy of Bahman Kalantari: Poly-
nomiography is the art and science of visualization in approxima-
tion of the zeros of polynomial equations using iteration functions.
Although its theoretical foundation can be traced to the well-
known Fundamental Theorem of Algebra, polynomiography offers
a new and exciting view into the world of polynomials as well
as the mysteries of this beautiful theorem itself. Not only is
polynomiography interesting and useful from the scientiﬁc and
educational points of view, but it turns the ancient root-ﬁnding
problem into a serious medium for creating artwork of great
variety and diversity through a combination of human creativity
and computer power. Each polynomial gives rise to an inﬁnite
number of 2D images, each called a polynomiograph. Each natural
number can be identiﬁed as a polynomial. Hence, for each poly-
nomial and each natural number there is an inﬁnite number of
polynomiographs waiting to be discovered. The particular image
on the cover is based on a polynomiograph produced by one of the
techniques in polynomiography, referred to as Voronoi coloring.
The title of the image is Acrobats. For more information, visit
This book has been composed in New Century Schoolbook
Printed on acid-free paper. ∞
Printed in the United States of America