Sacred Mathematics: Japanese Temple Geometry (review)

What kind of book is Sacred Mathematics? The authors intend it to be read in at least three ways: "as an art book that delights simply by the perusal of it, as a history book that provides a little insight into an aspect of Japanese culture rarely mentioned in standard surveys, and finally as a problem book that provides challenging exercises at both the high school and college levels" (p. xxi). In meeting these aims, Fukagawa Hidetoshi and Tony Rothman have crafted an excellent and well-thought-out volume. The figures, illustrations, and orthography of early modern Japanese works on mathematics are beautiful to look at, as are sangaku, wooden tablets recording mathematical problems posted in temples and shrines, the "temple geometry" of the title. With its shiny paper and copious supply of gorgeous illustrations, this is indeed a fine coffee-table book. As a work of history, the description "a little insight" is about right; no prior knowledge of Japanese history is assumed, and the cultural history does not get in the way of the mathematics. Finally, as a problem book, Sacred Mathematics excels; indeed the authors have been too modest: this volume is sure to prove an absolute delight for people who love mathematical puzzles, the modern equivalent of those of us who in the 1960s devoured the columns of Martin Gardner in Scientific American. It goes well beyond providing entertaining problems, for it also teaches important mathematical techniques. In addition, there is a nuanced and informed discussion of questions of precedence in mathematical discovery between Japan and the outside world. Still, for this forum, whose readers are probably not all lovers of mathematical puzzles, we must consider also what this book is not. It is not a work of academic research of the kind that MN readers are perhaps used to; in general, it lacks citations of primary sources and adequate surveys of secondary works, and, outside the purely mathematical, claims are made fairly freely. It does not, moreover, provide resources for the further study of Japanese mathematics (wasan). Finally, there is little serious attempt to discuss what mathematics was to the creators of the puzzles, nor to face the profound and important issues of cultural history that the topic of wasan raises.

The words on the page appear to be primarily those of Tony Rothman, a theoretical physicist and author of many works of science fiction, popular science, and scientific history. It seems that he and his Japanese coauthor, Fukagawa Hidetoshi, never met during the book's writing and production. As Rothman tells it, the bulk of the text was produced in imperfect English by Fukagawa and edited, sometimes substantially, by Rothman. It seems likely that Rothman actually contributed more than this implies; the book is highly readable, beautifully presented, and well organized, the polished product of a professional author. Rothman also apparently worked through all the problems, no mean feat. This is the third attempt by Fukagawa, an indefatigable student of wasan, to bring the subject of sangaku before the gaze of the English-speaking world, and it is the most splendid.

Before discussing the heart of the book, the collection of problems to be worked through, let us look at the surrounding elements. The first chapter gives us something of a potted history of wasan, including a survey of Japan in the Tokugawa period. It seems likely that we see Rothman's hand here and there; for example, a citation refers to Conrad Totman's Early Modern Japan. There is a bit of an effort to link the posting of sangaku in shrines and temples to "Shinto" attitudes, of which more later. By and large this chapter succeeds in providing context.

Filling out the context further, Sacred Mathematics offers recommendations for additional reading. These sections in the book are halfhearted and disappointing; an opportunity has been missed. While on this point I will mention some other things I would have liked to have seen in the book generally. Naturally...