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

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