The mathematics of Greece, Arabia, Egypt, Babylon, India and China, and their influence on modern maths, is widely documented. But the more recent maths of Japan, and its suitability as a source of ideas for use in schools, is less well known. During most of the Edo period (1603-1867), Japan cut itself off almost completely from the rest of the world. The effects of this policy were not entirely negative: cultural activities flourished, such as Noh and Kabuki theatre and the writing of haiku, and a distinctive type of maths appeared, known as wasan. This maths was not completely different from the maths of other cultures (since 6AD Japan had imported classical Chinese texts on arithmetic, algebra and geometry) but it developed along its own lines. Wasan results were originally displayed on wooden tablets called sangaku, containing text and beautifully coloured figures, hung under the roofs of shrines and temples; the oldest surviving tablet dates from 1683. Wasan problems are not just of historical interest. They display fascinating results that are not found elsewhere, and they often provide the challenge of finding simpler solutions than those originally printed. Some problems are very difficult to solve; others lead to generalisations and to modern geometrical research. Some can be used at various levels in schools, and for maths clubs and masterclasses.

Here are some examples, suitable for various levels of secondary school geometry.

In figure 1, find the radii of the various small circles and semicircles in terms of the radius of the large semicircle. This involves algebra and Pythagoras’s theorem, but once it has been discovered that the centres of the circles form various 3-4-5 right triangles, the way in which circles with radii 1, 2 and 3 fit inside a circle with radius 6 can easily be shown. Another problem involves showing that a 3-4-5 triangle has an incircle with radius 1.

In figure 2, a rectangle of paper is folded so that two opposite corners coincide. If the longer side of the rectangle is of given fixed length, what shape of rectangle gives the greatest value of the area ABC? This is a nice exercise in geometry and calculus.

In figure 3, a variable rhombus is inscribed in a fixed ellipse whose semi-axes have lengths a and b. Show that the radius r of its inscribed circle is fixed, and find r in terms of a and b. This is a good exercise involving co-ordinate geometry.

Four squares are connected as shown in figure 4. Find a relation between the side-lengths a, b, c and d of the squares, and show that the points P, Q and R lie on a line if and only if a = 2d.

One other aspect of wasan is that people apparently enjoyed performing complicated calculations on a soroban (abacus) as much as they enjoyed finding elegant proofs of elegant results. For instance, Yoshisuke Matsunaga (1693-1744) correctly calculated the first 51 digits of 9 with the aid of a soroban, using an infinite series.

John Rigby works at the School of Mathematics, Cardiff University, Cardiff CF24 4YH. E-mail: rigby@cardiff.ac.uk

FURTHER READING

Japenese Temple Geometry Problems by Hidetoshi Fukagawa and Dan Pedow. Chales Babbage Research Center, e-mail cbi@tc.umn.edu

www.cbi.umn.edu

Traditional Japenese Mathematics Problems of the 18th and 19th Centuries by Hidetoshi Fukagawa and John F Rigby. SCT Publishing, Singapore (forthcoming)

Japenese Temple Geometry by Tony Rotherman (with the co-operation of Hidetoshi Fukagawa), ‘Scientific American’, May 1998, pp84-9`.