## Sangaku: Reflections on the Phenomenon

Year after yearOn the monkey's face A monkey's mask. |

Matsuo Basho (1644-1694) |

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 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

*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

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.)

### References

- H. Fukagawa, D. Pedoe,
*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6 - H. Fukagawa, A. Rothman,
*Sacred Mathematics: Japanese Temple Geometry*, Princeton University Press, 2008 - J. Konhauser, D. Velleman, S. Wagon,
*Which Way Did the Bicycle Go?*, MAA, 1996, #50 - S. Roberts,
*King of Infinite Space*, Walker & Company, 2006 - D. E. Smith and Yoshio Mikami,
*A History of Japanese Mathematics*, Dover, 2004 (originally 1914)

## Sangaku

- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
^{th}Degree Challenge - A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

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