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.

Four squares are hinged as shown. When points A, B, C are collinear, what is the relationship between the sides of squares BEKH and KINS?


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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. Michel Cabart gave a solution that employed complex numbers. 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:

Draw the three dashed squares; and contemplate the figure in detail; the result is trivial.

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:

four hinged squares

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

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solution to four hinged squares sangaku

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:

second four hinged squares sangaku

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:

a² = b² + d² - 2bd·X
c² = b² + d² + 2bd·X,

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

Adding the two gives:

2d² = a² + c² - 2b²,

from which d is found to be

d = ½(a² + c² - 2b²).

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 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored 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
  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
  68. Another Sangaku in Square
  69. Sangaku via Peru
  70. FJG Capitan's Sangaku

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Copyright © 1996-2017 Alexander Bogomolny

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