Sacred Mathematics
Japanese Temple Geometry

by Fukagawa Hidetoshi and Tony Rothman

A certain ambiguity

A result of an unusual collaboration of two authors who never met, this is a glamorous book which will be treasured by all mathematics fans and especially by lovers of geometry.

The period that began in the early seventeenth century and lasted a little past the mid-nineteenth holds fascination for any student of Japanese history. During this period of roughly 200 years the country was almost entirely closed to foreign influence; travel to and from the West was banned and considered a capital offence. Trade with the West was channeled through the man-made miniature island of Deshima in Nagasaki harbor. (However, trade with China and Korea was not so obstructed.) Deshima was surrounded by a wall and joined to the mainland by a guarded bridge.

This period of seclusion - sakoku in Japanese - is also known as the "Great Peace". In the absence of warfare, samurai sought administrative jobs and gentlemen were supposed to cultivate skills in "medicine, poetry, the tea ceremony, music, ... arithmetic and calculation, ... reading and writing ..." Travel - mostly on foot - became secure and popular; many poets and mathematicians traveled throughout the country sight seeing, visiting temples and friends, and sharing their art and knowledge.

During that time, a strange custom arose of hanging wooden tablets with mathematical problems under the roofs of shrines and temples. These tablets are known in Japanese as sangaku, and we may only guess at their exact purpose. Some were probably a challenge to others, while some may have been offered to gods in gratitude for an inspiring problem. This explains both parts of the title Sacred Mathematics and Japanese Temple Geometry.

  a sample sangaku

The book presents much more than a gorgeous narrative of sangaku problems. The authors paint an extensive historic background of Japanese Art and Mathematics, which begins with the influence of Chinese mathematics and the introduction of abacus to the islands. The narrative is enhanced with biographies of many contemporary mathematicians and an outline of their work, and includes a chapter of extracts from the travel diary of the 19th century mathematician Yamaguchi Kanzan. Solutions are provided for most of the problems in the book, both simple and complex, and an additional chapter carries a comparison with similar problems published in the West.

One other collection of sangaku is available in the English speaking world. This was published by Dan Pedoe and H. Fukagawa some 20 years ago and has long since become a bibliographic rarity. It may be found on the Web for close to $100. The present volume not only transcends the former in quality and breadth of material, but brings out a peculiar Japanese cultural phenomenon to a much broader audience at an affordable price of $35 ($23.45 with a discount.) The book will be of value to teachers of geometry and history of mathematics; fans of mathematics will find it especially enjoyable.

Sakoku was a remarkable period in the history of Japan, but no less remarkable was the manner in which it ended and the repercussions its end had on the development of Japanese society in general and on mathematics in Japan in particular. On July 8, 1853 an armada of four steamships commanded by Commodore Matthew Perry anchored at Edo (Tokyo) Bay. The impression left by the ships' great guns and novel Western technologies was of mystical proportions. Nearly one year later, in March 1854, when Perry returned at the head of seven steamships, a peace treaty was signed between the US and Japan that effectively opened Japan to Western trade and international cooperation. Similar treaties were soon signed with other Western countries, and Japan entered a period of modernization. The practice of hanging sangaku tapered out, as Western science and mathematics supplanted the Japanese tradition.

The effect was drastic. At the end of the ICME-9 that took place in Makuhari near Tokyo in the year 2000, the participants toured the temples and shrines in the Kyoto area, but the guides never found it necessary to draw our attention to any sangaku or traditional Japanese mathematics. In the Preface, Dr. Fukagawa Hidetoshi, a high school math teacher, describes his first encounter with sangaku in 1969 when a teacher of traditional Japanese literature asked him to decipher an 1815 book printed from wooden blocks. This suggests that in the 20th century wasan - the traditional Japanese mathematics, which flourished during the Edo period - was all but forgotten.

Fukagawa Hidetoshi, one of the authors of the book, is chiefly responsible for the revival of the interest in wasan. And this, in particular, explains the great enthusiasm with which the book was created. The outcome is a wonderful, lavishly illustrated, exquisite work of art and mathematics, a worthy tribute to the charming beauty and peculiar ingenuity of the mathematical tablets.

My one reservation about the book is a mental block of sorts. At one point, the authors' enthusiasm for the sangaku phenomenon carried them away towards an unjustified and implausible claim. Even allowing for a significant dose of literary license the authors' estimate of sangaku's popularity seems exaggerated. We read (p. 1):

  The creators of these sangaku - a word that literally means "mathematical tablet" - hung them by the thousands in Buddhist temples and Shinto shrines throughout Japan ...

and later (pp. 9-10):

  ... the inscriptions on the tablets make clear that whole classes of students, children, and occasionally women dedicated sangaku. So the best answer to the question "Who created them?" seems to be "everybody."

Realistically, this is impossible. The number of surviving tablets is about 900, and according to the 1721 census, the population of Japan was approximately 30 million, including 4 million samurai families and their attendants. The population of Edo (the future Tokyo) was approximately 1 million, and that of Kyoto 400,000. Now, this historic period lasted for roughly 200 years. Do the arithmetic. The authors themselves write in one place:

  So it is reasonable to guess that there were originally thousands more than the 900 tablets extant today.

If by "thousands" the authors mean "a few thousands", this is an estimate I am willing to accept.

Sacred Mathematics - Japanese Temple Geometry, Fukagawa Hidetoshi, Tony Rothman, Princeton University Press, 2008


  1. Sangaku: Reflections on the Phenomenon
  2. Critique of My View and a Response
  3. 1 + 27 = 12 + 16 Sangaku
  4. 3-4-5 Triangle by a Kid
  5. 7 = 2 + 5 Sangaku
  6. A 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored Sangaku Problem
  10. A Sangaku: Two Unrelated Circles
  11. A Sangaku by a Teen
  12. A Sangaku Follow-Up on an Archimedes' Lemma
  13. A Sangaku with an Egyptian Attachment
  14. A Sangaku with Many Circles and Some
  15. A Sushi Morsel
  16. An Old Japanese Theorem
  17. Archimedes Twins in the Edo Period
  18. Arithmetic Mean Sangaku
  19. Bottema Shatters Japan's Seclusion
  20. Chain of Circles on a Chord
  21. Circles and Semicircles in Rectangle
  22. Circles in a Circular Segment
  23. Circles Lined on the Legs of a Right Triangle
  24. Equal Incircles Theorem
  25. Equilateral Triangle, Straight Line and Tangent Circles
  26. Equilateral Triangles and Incircles in a Square
  27. Five Incircles in a Square
  28. Four Hinged Squares
  29. Four Incircles in Equilateral Triangle
  30. Gion Shrine Problem
  31. Harmonic Mean Sangaku
  32. Heron's Problem
  33. In the Wasan Spirit
  34. Incenters in Cyclic Quadrilateral
  35. Japanese Art and Mathematics
  36. Malfatti's Problem
  37. Maximal Properties of the Pythagorean Relation
  38. Neuberg Sangaku
  39. Out of Pentagon Sangaku
  40. Peacock Tail Sangaku
  41. Pentagon Proportions Sangaku
  42. Proportions in Square
  43. Pythagoras and Vecten Break Japan's Isolation
  44. Radius of a Circle by Paper Folding
  45. Review of Sacred Mathematics
  46. Sangaku à la V. Thebault
  47. Sangaku and The Egyptian Triangle
  48. Sangaku in a Square
  49. Sangaku Iterations, Is it Wasan?
  50. Sangaku with 8 Circles
  51. Sangaku with Angle between a Tangent and a Chord
  52. Sangaku with Quadratic Optimization
  53. Sangaku with Three Mixtilinear Circles
  54. Sangaku with Versines
  55. Sangakus with a Mixtilinear Circle
  56. Sequences of Touching Circles
  57. Square and Circle in a Gothic Cupola
  58. Steiner's Sangaku
  59. Tangent Circles and an Isosceles Triangle
  60. The Squinting Eyes Theorem
  61. Three Incircles In a Right Triangle
  62. Three Squares and Two Ellipses
  63. Three Tangent Circles Sangaku
  64. Triangles, Squares and Areas from Temple Geometry
  65. Two Arbelos, Two Chains
  66. Two Circles in an Angle
  67. Two Sangaku with Equal Incircles
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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