An enchanting history of Japanese geometry--of a time and place where 'geometers did not cede place to poets.' This intersection of science and culture, of the mathematical, the artistic, and the spiritual, is packed, like circles within circles, with rewarding Aha! epiphanies that drive a mathematician's curiosity.

Teachers will welcome this remarkable collection of mathematical problems, history, and art, which will enrich their curriculum and promote both logical thinking and critical evaluation. It is especially important that we maintain an interest in geometry, which needs, and for once gets, more than its share.

This remarkable book provides a novel insight into the Japanese mathematics of the past few hundred years. It is fascinating to see the difference in mathematical style from that which we are used to in the Western world, but the book also elegantly illustrates the cross-cultural Platonic nature and profound beauty of mathematics itself.

A significant contribution to the history of mathematics. The wealth of mathematical problems--from the very simple to quite complex ones--will keep the interested reader busy for years. And the beautiful illustrations make this book a work of art as much as of science. Destined to become a classic!

A pleasure to read. brings to light the unique style and character of geometry in the traditional Japanese sources--in particular the problems. These problems range from trivial to utterly devilish. I found myself captivated by them, and regularly astounded by the ingenuity and sophistication of many of the traditional solutions.

Between the seventeenth and nineteenth centuries Japan was totally isolated from the West by imperial decree. During that time, a unique brand of homegrown mathematics flourished, one that was completely uninfluenced by developments in Western mathematics. People from all walks of life--samurai, farmers, and merchants--inscribed a wide variety of geometry problems on wooden tablets called sangaku and hung them in Buddhist temples and Shinto shrines throughout Japan. Sacred Mathematics is the first book published in the West to fully examine this tantalizing--and incredibly beautiful--mathematical tradition.

Fukagawa Hidetoshi and Tony Rothman present for the first time in English excerpts from the travel diary of a nineteenth-century Japanese mathematician, Yamaguchi Kanzan, who journeyed on foot throughout Japan to collect temple geometry problems. The authors set this fascinating travel narrative--and almost everything else that is known about temple geometry--within the broader cultural and historical context of the period. They explain the sacred and devotional aspects of sangaku, and reveal how Japanese folk mathematicians discovered many well-known theorems independently of mathematicians in the West--and in some cases much earlier. The book is generously illustrated with photographs of the tablets and stunning artwork of the period. Then there are the geometry problems themselves, nearly two hundred of them, fully illustrated and ranging from the utterly simple to the virtually impossible. Solutions for most are provided.

A unique book in every respect, Sacred Mathematics demonstrates how mathematical thinking can vary by culture yet transcend cultural andgeographic boundaries.

Fukagawa Hidetoshi is a retired high-school teacher in Japan and one of the world's experts on "sangaku". He is the coauthor of "Japanese Temple Geometry Problems". Tony Rothman is a theoretical cosmologist who lectures in physics at Princeton University. His books include "Everything's Relative and Other Fables from Science and Technology".

Foreword by Freeman Dyson ix

Preface by Fukagawa Hidetoshi xiii

Preface by Tony Rothman xv

Acknowledg ments xix

What Do I Need to Know to Read This Book? xxi

Notation xxv

Chapter 1: Japan and Temple Geometry 1

Chapter 2: The Chinese Foundation of Japanese Mathematics 27

Chapter 3: Japa nese Mathematics and Mathematicians

of the Edo Period 59

Chapter 4: Easier Temple Geometry Problems 89

Chapter 5: Harder Temple Geometry Problems 145

Chapter 6: Still Harder Temple Geometry Problems 191

Chapter 7: The Travel Diary of Mathematician Yamaguchi Kanzan 243

Chapter 8: East and West 283

Chapter 9: The Mysterious Enri 301

Chapter 10: Introduction to Inversion 313

For Further Reading 337

Index 341