In the Wasan Spirit

As Tony Rothman explained in his trend-setting article,

There is a word in Japanese, wasan, that is used to refer to native Japanese mathematics. Wasan is meant to stand in opposition to yosan, or Western mathematics. ... To the extent that it makes sense to credit anyone with the founding of wasan, that honor probably goes to Mori and Yoshida (1598-- 1672). ... Wasan, though, was created not so much by a few individuals but by something much larger."

And later,

... by the next century, books were being published that contained typical native Japanese problems: circles within triangles, spheres within pyramids, ellipsoids surrounding spheres. The problems found in these books do not differ in any important way from those found on the tablets, and it is difficult to avoid the conclusion that the peculiar flavor of all wasan problems--including the sangaku--is a direct result of the policy of national seclusion.

I do not share Rothman's attribution of nested figures as "native Japanese" (remember the one on Archimedes' tombstone?) or related to the policy of seclusion. (Why not to ascribe the development of projective geometry by J.-V. Poncelet to his longing for the distant France while in Russian captivity?) I also disagree with Rothman's assessment of sangaku popularity. Nonetheless, there are indeed several sangaku dealing with various configurations of tangent circles. Below I discuss a couple of simple ones that appear as likely candidate for the attention of a budding sangaku devotee. One, from P. Yiu's article where he mentions the book by Fukagawa and Pedoe as a possible source, and the other occurred to me through a memory malfunction when I tried to elicit a recollection of P. Yiu's example. Both problems are easily solved with two applications of the Pythagorean theorem.

Problem 1

The centers A and B of two circles lie on the other circle. Construct a circle tangent to the line AB, to the circle (A) internally, and to the circle (B) externally.

Solution

Problem 2

A circle is tangent internally to a bigger circle and its diameter. Construct the circle tangent to both and to that diameter and express its radius in terms of the large circle.

Solution

References

1. T. Rothman, Japanese Temple Geometry, Scientific American, May, 1998
2. 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

3. P. Yiu, Elegant Geometric Constructions, Forum Geometricorum, 5 (2005), pp. 75-96

Sangaku

Solution to Problem 1

Based on the diagram, let AB = a, AF = x, and OF = r (the radius of the sought circle.) Then in ΔBFO,

(a + r)2 = r2 + (a + x)2.

And in ΔAFO,

(a - r)2 = r2 + x2.

Subtraction gives

4ar = a2 + 2ax,

or

x + a/2 = 2r,

which means that side EF of the square MCEF, M being the midpoint of AB, is a diameter of the sought circle. The construction is now easy.

Solution to Problem 2

Based on the diagram below, let R be the radius of the middle circle (E), r the unknown radius of the circle in question.

(R + r)2 = AD2 + (R - r)2,

or

In ΔABC,

(2R - r)2 = BC2 + r2,

or

R2 - 4Rr = BC2.

Since AD = BC, we have

4Rr = 4R2 - 4Rr,

or R = 2r. Center A of the circle can then be found at the intersection of circles with radii 3R/2 centered at E and C.