As the reader may notice, Israel Gelfand is the person who most
influenced my outlook on mathematics. I am most grateful to him
for generously sharing with me his ideas and incisive observations.
I am indebted to Gregory Cherlin and Reuben Hersh and to my
old friend Owl for most stimulating conversations and many com-
ments on the book; some of the topics in the book were included on
Almost everyday chats with Hovik Khudaverdyan about math-
ematics and the teaching of mathematics seriously contributed to
my desire to proceed with this project.
During our conversation in Paris, the late Paul Moszkowski
forcefully put forth the case for the development of the theory of
Coxeter groups without reference to geometry and pointed me to-
ward his remarkable paper .
Jeff Burdges, Gregory Cherlin, David Corfield, Chandler Davis,
Ed Dubinsky, Erich
David Pierce, John Stillwell, Robert Thomas, Ijon Tichy, and Neil
White carefully read and corrected the whole or parts of the book.
My thanks are due to a number of people for their advice and
comments on the specific areas touched upon in the book: to David
Corfield—on the philosophy of mathematics, to Susan Blackmore—
on memetics, to Vladimir Radzivilovsky—for explaining to me the
details of his teaching method, to Satyan Devadoss—on diagrams
and drawings used in this book, to Ray Hill—on the history of
coding theory, to Peter
universal algebra, to Sergey
Utyuzhnikov—on chess, turbulence, and dimensional analysis, to
Alexander Jones and Jeremy Gray—on the history of Euclidean ge-
ometry, to Victor Goryunov—on multivalued analytic functions, to
Thomas Hull—on the history of Origami, to Gordon Royle—on Su-
doku, to Alexander Kuzminykh and Igor Pak—on convex geometry,
to Dennis Lomas—on visual thinking,
philosophy and convex geometry, and, finally, to Paul Ernest and
Inna Korchagina for general encouraging comments.
Jody Azzouni, Barbara Sarnecka, and Robert Thomas sent me
the texts of their papers [5, 6], [163, 225], .
David Petty provided diagrammatic instructions for the Origami
Chinese Junk (Figures 11.2 and 11.3). Dougald Dunham allowed
me to use his studies of hyperbolic tessellations
engravings (Figures 5.4 and 5.5). Ali Nesin made illustrations for
Chapter 10. Simon Thomas provided me with diagrams used in
I am lucky that my university colleagues David Broomhead,
Paul Glendinning, Bill Lionheart, and Mark Muldoon are involved
in research in mathematical imaging and/or mathematical models
of neural activity and perception; their advice has been invaluable.
to Semen Kutateldaze—on
Ray Hill, Tony Gardiner, Chris Hobbs,