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.

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!

References

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)

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