Sangaku: Reflections on the Phenomenon

Year after year
On the monkey's face
A monkey's mask.
Matsuo Basho

Sangaku are often colorful tablets offered in shinto shrines (and sometimes in buddhist temples) in Japan and posing mathematical problems. The earliest sangaku date a few years before the beginning of the japanese Edo period (1603-1867) of self-imposed seclusion from the Western world. Most of the write-ups on the Sangaku phenomenon are based on either a Scientific American article by Tony Rothman written in co-operation with Hidetoshi Fukagawa, a Japanese teacher with a Ph.D in mathematics, or the book by H. Fukagawa and D. Pedoe. For example, Rothman explains in the introduction to his article:

Of the world's countless customs and traditions, perhaps none is as elegant, nor as beautiful, as the tradition of sangaku, Japanese temple geometry. From 1639 to 1854, Japan lived in strict, self-imposed isolation from the West. Access to all forms of occidental culture was suppressed, and the influx of Western scientific ideas was effectively curtailed. During this period of seclusion, a kind of native mathematics flourished.

Devotees of math, evidently samurai, merchants and farmers, would solve a wide variety of geometry problems, inscribe their efforts in delicately colored wooden tablets and hang the works under the roofs of religious buildings. These sangaku, a word that literally means mathematical tablet, may have been acts of homage--a thanks to a guiding spirit--or they may have been brazen challenges to other worshipers: Solve this one if you can! For the most part, sangaku deal with ordinary Euclidean geometry. But the problems are strikingly different from those found in a typical high school geometry course. Circles and ellipses play a far more prominent role than in Western problems: circles within ellipses, ellipses within circles. Some of the exercises are quite simple and could be solved by first-year students. Others are nearly impossible, and modern geometers invariably tackle them with advanced methods, including calculus and affine transformations.

Introducing a Sangoku problem, the authors of a remarkable problem collection Which Way Did the Bicycle Go? quote from the book by H. Fukagawa and D. Pedoe:

During the greater part of the Edo period (1603-1867) Japan was almost completely cut off from the western world. Books on mathematics, if they entered Japan at all, must have been scarce, and yet, during this long period of isolation people of all social classes, from farmers to samurai, produced theorems in Euclidean geometry which are remarkably different from those produced in the west during the centuries of schism, and sometimes anticipated these theorems by many years.

These theorems were not published in books, but appeared as beautifully coloured drawings on wooden tablets which were hung under the roof in the precincts of a shrine or temple.

Remarkably, where Rothman notes the difference between Sangaku and school geometry problems, Pedoe contrasts Sangaku with the geometry "produced in the west". In general, I believe, the tendency to exaggerate the significance of the Sangaku phenomenon grows with the follow-up writings. For example, Chad Boutin starts his paper in News@Princeton thus

Perhaps it's not surprising that sudoku - the number puzzles that everyone seems to be working on these days - first became popular in Japan before spreading across the ocean. The fad is reminiscent of a math craze that swept the islands centuries ago, when ardent enthusiasts went so far as to turn the most beautiful geometrical solutions into finely illustrated wooden tablets, called sangaku, that adorned the walls of local temples and shrines.

Before I proceed, I wish to assure the reader of my deep appreciation of the quality and beauty of many Sangaku problems. At the bottom of this page there is a list of problems discussed at this site. I just question the plausibility of the historic picture that emerges from the above quotes. Essentially, I doubt that the practice was widely spread or that it was much affected by the atmosphere of seclusion. It perhaps relevant to note that during this same period Haiku became a national activity. The commonly given reason for the growth of Haiku's stature in Japanese culture is the involvement of two masters (Matsuo Basho and Onitsura) who elevated haiku to new artistic heights. Never did I see the ascendency of Haiku related to the particulars of the period, although, in all likelihood, it was. By limiting the influx of intellectual stimuli, Sakoku, as this period of self-imposed seclusion is known in Japanese, may have caused a more focused concentration on the homegrown developments.

First of all, Sangaku problems were written in Kanbun, Chinese written for the Japanese audience. While Japan may boast high level of literacy even during the Edo period, Kanbun is said to be the Japanese equivalent of Latin, with the obvious implication. As [Fukagawa & Pedoe, Preface] note, ... Kanbun ... can be read by only a small number of people in modern Japan. [Fukagawa and Rothman, p. 9] note:

Sangaku are inscribed in a language called Kanbun, which used Chinese characters and essentially Chinese grammar, but included diacritical marks to indicate Japanese meaning. Kanbun played a role similar to Latin in the West and its use on sangaku would indicate that whoever set down the problems was highly educated. The majority of the presenters, in fact, seem to have been members of the samurai class.

This makes it hard to believe that Kanbun was widely known during the Edo period, either.

Second, Sakoku lasted a little more than 200 years, from 1647 through 1854. (The sangaku tradition has started earlier and ended later so that I estimate its duration at 250 years.) Europe, during the first one hundred years of Sakoku, saw the emergence of Calculus but not much of geometry, except, perhaps for a few Euler's results. In Europe, in the second half of Sakoku, numerous cases have been documented (e.g., Wessel's contribution to complex numbers) when important mathematical results have been overlooked by the mathematical community or have been independently rediscovered in different corners of the continent. So waxing sentimental over "the centuries of schism" is not quite justified. If anything, proliferation of Sangaku demonstrates the effectiveness of popularization. Reaching the broad masses via beautiful and mysterious wooden tablets displayed at places of worship and congregation, mathematics got a foothold in Japanese culture. But even this did not last long. By the 20th century, the Sangaku tradition was all but forgotten.

In her biography of H. M. S. Coxeter, Siobhan Roberts quotes her correspondence (2003-2005) with Koji Miyazaki, the author of 35 books on various aspects of geometry. "Us Japanese originally don't like geometric logic and if there are no Coxeter's understandable geometry us Japanese have not become aware of morphologic contents about things. Japanese version of his book 'Introduction into Geometry' gave much impact to so many Japanese including me."

Third, it is also said that

During the period, Japan progressively studied Western sciences and techniques (called rangaku, literally "Dutch studies") through the information and books received through the Dutch traders in Dejima. The main areas that were studied included geography, medicine, natural sciences, astronomy, art, languages, physical sciences such as the study of electrical phenomena, and mechanical sciences as exemplified by the development of Japanese clockwatches, or wadokei, inspired from Western techniques.

I read from here that the Japanese did not lose much, at least not for the first 100 years of seclusion.

Fourth, another problem with the quotes is that during the Edo period, the population was divided into four classes: the samurai on top (about 5% of the population), and the peasants (more than 80% of the population) on the second level. Below the peasants were the craftsmen, and even below them, on the fourth level, were the merchants. So, perhaps, the expression "... people of all social classes, from farmers to samurai, produced theorems ..." is not quite accurate and does not probably express correctly the authors intention to hint at the broad popularity of Sangaku. It certainly does not embrace all the strata of the contemporary Japanese society.

Fifth, by the mid-eighteenth century, population of Edo - the future Tokyo - reached 1,000,000. That of Kyoto and Osaka was around 400,000 each. On the other hand, the highest number of surviving Sangaku I came across is given as 880. On some the problems are barely visible. If one did not know better, they could be mistaken for the plain wooden pieces. I think an estimate of a total of 5000 produced during the period of seclusion may be quite plausible if not exaggerated. It appears that, on average, during 250 years of Sakoku about 20 Sangaku have been created per year. Which looks as a possible outcome of a small group of disciples rather than the broad population as seen implied by the quote ("people of all social classes".)

A later correction: Chapter 7 of the new book by Fukagawa and Rothman narrates the story of Kazan Yamaguchi, a Japanese mathematician who undertook six "sangaku pilgrimages" visiting the temples around the country over the period between 1817 and 1828. Yamaguchi's diary that runs about 700 pages documents 87 sangaku of which only 2 have survived to the present. Using the survival factor of 87/2 the above calculations lead to 880·87/2/250 ≈ 155 tablets per year. This still could be accounted for by a few schools of traveling mathematicians. Also, putting down the Descriptions of only 87 sangaku seems like a particularly meager outcome of 6 travels around the country.

Sixth, many of the surviving Sangaku come in groups of related problems, see for example, Sequences of Touching Circles, Sangakus with a Mixtilinear Circle or Circles in a Circular Segment. ([Fukagawa & Pedoe, Preface] also suggest that, towards the end of seclusion, there was undoubted plagiarism.) Related problems appeared in different prefectures, not just in different shrines. Which serves to indicate that much of the Sangaku proliferation was due to travel, either by a peripatetic school or wandering individuals.

The foregoing thoughts should not be construed as the lack of appreciation of the Sangaku phenomenon or of particular problems. Many of the problems I have seen have charming elegance. I just do not feel they need any kind of extraneous mystification. They deserve full appreciation simply for what they are.

In a 1914 book, D. E. Smith and Y. Mikami mention temple geometry in a more level headed manner (p. 184):

Fujita's son Fujita Kagen (1765-1821) was also a mathematician of prominence. He published in 1790 his Simpeki Sampo (Mathematical problems Suspended before the Temple), and in 1806 a sequel, the Zoku Simpeki Sampo. The significance of the name is seen in the fact that the work contains a collection of problems that had been hung before various temples by certain mathematical devotees between 1767 and the time when Fujita wrote, together with the rules for their solution. This strange custom of hanging problems before the temples originated in the seventeenth century, and continued for more than two hundred years. It may have arisen from the desire for praise or approval of the gods, or from the fact that this was a convenient means of publishing a discovery, or from the wish to challenge others to solve a problem, as European students in the Middle ages would post a thesis on the door of a church.

(Noteworthy: Jean Constant, a contemporary artist, has been inspired by the tablets to create many wonderful artwork he collected in several online galleries.)


  1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

    Write to:

    Charles Babbage Research Center
    P.O. Box 272, St. Norbert Postal Station
    Winnipeg, MB
    Canada R3V 1L6

  2. H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008
  3. J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996, #50
  4. S. Roberts, King of Infinite Space, Walker & Company, 2006
  5. D. E. Smith and Yoshio Mikami, A History of Japanese Mathematics, Dover, 2004 (originally 1914)


  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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