Four Hinged Squares

The purpose of the applet below is to illustrate an 1826 sangaku hung by Ikeda Sadakazu in an Azabu shrine, Tokyo. The tablet since disappeared but not before it was recorded in an 1827 book Shamei Sanpu (Sacred Mathematics) by Shiraishi Nagatada (1795-1862). The problem has also been included in an 1840 collection Sanpo Chokujutsu Seikai (Mathematics without Proof) by Heinouchi Masaomi.

I first stumbled upon this simple problem in the Temple Geometry by Fukagawa and Pedoe, where it was listed without solution. My solution that employs Bottema's theorem appears on a separate page. The newer book Sacred Geometry by Fukagawa and Rothman also lists the problem but now with the original solution from Sanpo Chokujutsu Seikai where it read:

Fukagawa and Rothman give a solution nonetheless.

Shamei Sanpu in general contains by far more difficult problems. For example, the page where the four hinged squares is presented also includes a problem of determining the surface area of an ellipsoid:

References

1. 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
   Canada R3V 1L6

2. H. Fukagawa, A. Rothman, Sacred Geometry: Japanese Temple Geometry, Princeton University Press, 2008, p. 149

Copyright © 1996-2008 Alexander Bogomolny

Square BEKH is inscribed in square UVTR so that

HV = KT

(and also EU = KR.) In the square circumscribing square CYIH,

HV = IQ

implying KT = IQ. Let M be the intersection of QT and IK. Then triangles KTM and IQM are congruent so that M is the midpoint of IK (KM = IM). Similarly, on the other side, KL = SL.

We conclude that since KM = KL and angle LKM is right while angle RKT is straight, triangles LRK and KTM are congruent. In square UVTR, KR = HT. So it follows that triangles LRK and KTH are also congruent leading to

Δ KTH = Δ KTM

and KM = KH. Just what was needed.

Incidently, there is another sangaku with a similar configuration:

This has been written in 1832 also in the Tokyo prefecture and was recorded in Kokon Sankan by Uchida Gokan [Temple Geometry, p. 131]. The problem is to express d in terms of a, b, c. The solution is more conventional. Apply the Law of Cosines to triangles abd and bcd:

where X is the cosine of the angle between b and d in triangle abd.

Adding the two gives:

from which d is found to be

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

Copyright © 1996-2008 Alexander Bogomolny