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

Copyright © 1996-2009 Alexander Bogomolny