#2           58. A CHINESE PUZZLE                  

The Monad is a symbol of particular religious significance in China.

The black and white parts are known as the Yin and Yang, and oriental scholars interpret the symbol in terms of the male and female forces of nature. It is also said that the two piece cover of a basball was inspired by this symbol!

We have two problems. The first is to use one continuous line to divide the Yin and Yang so that the circle is divided into four pieces having the same size and shape.

The second, and a little more difficult, is to divide the Yin and Yang by a single straight line, so that each is cut into two parts of equal area.

PROBLEM 1 HINT

Since the the constructed pieces will all lie within the given circle, rotation is suggested.
PROBLEM 2 HINT

Since we want to divide the Yin and Yan evenly, a first try might be the diagonal line XY at an angle of 45°. If this were a correct choice, regions A and B would have to have the same area.
SOLUTION 1.
SOLUTION 2.

Draw diagonal diameter XY as shown, at an angle of 45°. This line will give the solution, providing regions A and B have the same area, Let the large circle have radius 1. Then semicircle A has diameter 1 and area /8. A quadrant of the large circle has area /4, so region B has area /4 – /8 = /8 as required.

EXTENSION

1. The monad has an interesting structure. In fact it is based on a circle with two largest inscribed circles meeting at the centre. If you do a similar construction with squares, you obtain a swastika. What happens with other shapes?

2. Try dividing the monad into 6, 8, ... congruent pieces. This is a good artistic exercise. Is there any interesting mathematics?