Japanese Art and Mathematics

As [Mikami and Smith, p. 279] wrote, "... the mathematics of Japan was like her art, exquisite rather than grand." In some instances Japanese mathematicians came up with problems years before identical problems have been solved in the West, see, e.g., Neuberg Sangaku or Malfatti's Problem.

However, "[they] never developed a great theory that in any way compares with the calculus as it existed when Cauchy, for example, had finished with it."

But exquisite it was and the illustrations that survive appeal to the timeless sense of beautiful. A wealth of material has been gathered in a superbly illustrated book Sacred Mathematics: Japanese Temple Geometry by H. Fukagawa and Tony Rothman. Although the primary subject of the book is Temple Geometry, the authors go an extra mile painting a vivid picture of the Japanese mathematical landscape in the Edo period. Much of contemporary mathematics has been presented in a peculiar format of wooden tablets - sangaku - hung in temples and shrines all over the country. But mathematics has been also spread by a more conventional means, via books and manuscripts and by itinerant teachers. In the first three chapters of their book, Fukagawa and Rothman give an eclectic account of the history of Japanese mathematics and of works by native mathematicians.

Here are just two examples from the book.

One illustration is plucked from the Chinese Suanfa Tong Zong, or Systematic Treatise on Mathematics by Cheng Da-wei, Chapter Excess and Lack (1592); the book that has a profound influence on traditional Japanese mathematicians.

Two reeds of equal height project 3 syaku above the surface of a pond. If we draw the top of one reed 9 syaku in the direction of the shore so that the top is just touching the surface of the water, find the depth of the pond.

Another examples deals with a practical problem of weighing an elephant. In the 1778 Funki Jinko-Ki, or Riches of Jinko-ki, an anonymous author suggests an approach that would without doubt be appreciated by the great Archimedes as much as by the modern health conscious public. Bring the elephant onto a boat and mark the water line. Remove the elephant, then fill the boat with stones of known weight until the water reaches the same level it did with the elephant. Quite an aerobic exercise is this!


  1. H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008
  2. D. E. Smith and Yoshio Mikami, A History of Japanese Mathematics, Dover, 2004 (originally 1914)


  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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