Sangaku Iterations
Is it Wasan?

I happened on a sangaku problem posed by the Japanese mathematician Tumugu Sakuma (1819-1896) who worked at the later years of the Edu period of seclusion by sheer accident. An article in Mathematics magazine by Fukuzo Suzuki just followed the one on the Lights Out puzzle I've been reading. The author referred to a problem in a Japanese book People of Wasan on Record by A. Hirayama (1965) and noted some incorrect results. The problem in the article was a generalization of that in Hirayama's book and asked to find certain relationships in a configuration of an equilateral triangle whose side lines passed though the vertices of a given isosceles triangle. (In the book, the original sangaku required a right isosceles triangle.)

Somehow, I found both the problem and the solution unappealing. However, the problem did not fit the stereotype of the sangaku promoted by Tony Rothman, whose article in Scientific American caused much stir in the math education community. The problem did not have "circles within triangles, spheres within pyramids, ellipsoids surrounding spheres." For this reason alone I thought it worthy to be included in my collection.

But how does one construct an equilateral triangle with the side lines through the vertices of another triangle? A recollection flashed through my mind of another problem where a triangle was obtained as a limit of an iterative procedure. This was a trivial matter to modify the applet and the result is below.

For a given triangle, you can start iterations anywhere by clicking a mouse button. On each step, the iterations go from a point in a direction of a vertex, using all three vertices in a loop. If p₀ is the starting point and v₀ the first vertex, then the second iterate is chosen according to the formula

The secondd is computed analogously via

The subsequent iterates are calculated by the formula that forces equal sides at the limit:

Why (1) leads to an equilateral triangle? Assuming the iterations converge, i.e.,

and using a = B₀C₀, b = A₀C₀, and c = A₀B₀, (1) gives in the limit

This is a system of three linear equations with three quantities a, b, c and solutions that, because of the symmetry, are bound to satisfy a = b = c.

References

1. F. Suzuki, An Equilateral Triangle with Sides through the Vertices of an Isosceles Triangle, Mathematics Magazine, Vol. 74, No. 4. (Oct., 2001), pp. 304-310.

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

Copyright © 1996-2009 Alexander Bogomolny