Pappus' Sangaku
Few sangaku illustrate better the disparity of the independent development of mathematics during the Edo period (16031867) of selfimposed seclusion of Japan from the Western world than those related to arbelos and Pappus' chain of circles. The tool of inversion so useful in solving problems involving circles remained unkown to the Japanese until late 19 century.
Consider the sangaku 1.8.2 from the collection by Fukagawa and Pedoe:
Circles C_{1}(r) and C_{2}(r) touch at O in the circle O(2r) and also touch O(2r), C_{1}(r) touching at T. O_{1}(r_{1}) touches O(2r) internally and both C_{1}(r) and C_{2}(r) externally, O_{2}(r_{2}) touches O_{1}(r_{1}) and C_{1}(r) externally and O(2r) internally, and so on. Find r_{n}.
References
H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
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Circles C_{1}(r) and C_{2}(r) touch at O in the circle O(2r) and also touch O(2r), C_{1}(r) touching at T. O_{1}(r_{1}) touches O(2r) internally and both C_{1}(r) and C_{2}(r) externally, O_{2}(r_{2}) touches O_{1}(r_{1}) and C_{1}(r) externally and O(2r) internally, and so on. Find r_{n}.
With the help of inversion the solution is immediate. In the generalization of Archimedes' formula
r_{n} = rk / (k²n² + k + 1).
where k is the ratio between the radii of the two inner circles, i.e.
r_{n} = r / (n² + 2),
exactly the expression that appears in the Gothic Arc diagram.
In the very first volume (1826) of Crelle's Journal, J. Steiner proved several theorem concerning arbelos, the above being one of them. The sangaku however is dated 1801 (and was written in the Mie prefecture.)
Sangaku

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Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny