Sangaku: Reflections on the Phenomenon

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:

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:

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

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:

This make 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 20^th 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

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):

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

References

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)

Sangaku

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

Copyright © 1996-2008 Alexander Bogomolny