YM COLUMN ARCHIVE

October 22, 2008

On ratio of the circumference of a circle to its diameter

Circumference ratio π is a precious number. Of course, its first reason is represented in such formulas about circle like “circumferential length = 2 x π x radius” or “area of circle = π x radius x radius”. It is amazing that surface area of sphere or its volume are easily calculated by using π. But its usefulness is not only about circle or sphere but also in many other fields of mathematics to make you surprised, “why is π in here?”

For example, π is most useful in the field of probability theory. Either because probability theory is closely connected with economic activity of the world or simply because of brilliant mind of Friedrich Gauss (1777~1855), 10 Deutsche Mark before adoption of Euro was dedicated in honor of Friedrich Gauss.

http://www-personal.umich.edu/~jbourj/images/money/gauss12.jpg

When you click the above URL, you will see normal distribution curve like a temple-bell to the right of Gauss and careful study of the note will discover π in the definition inside the bell.

To the best of my memory, perfume “Givency” has a brand by the name of π, and I remember π was used in the title of movie. Also, there is a social club for ladies and gentlemen called Spacelab “π”, which may bring you an imagination to watch movie π with perfume π on at club π. There is no other popular character in mathematics like π.

Over ten years ago, I met with a Japanese gentleman by the name Mr. Tomoyori (Sony) who is in the Guinness Book of Records for his memorizing π to 20,000 digits. I asked him,“Who was a record holder before you?” He answered “It’s an Indian to 10,000 digits.” “What is he doing now?” I asked again. “He is taking a hard training in Canada for 30,000 digits”. In fact, there is no end to the digits of π. And now, the question is how people calculated the ratio in earlier times when there was not even a handy calculator.

• They seemed to have depended on the method to use polygons inscribed in or circumscribed about a circle.

http://www004.upp.so-net.ne.jp/s_honma/circle/circle410.gif

To explain by the above URL figure, outer hexagon is longer than circle in length and inner hexagon is shorter than circle in circumference. The value of the circumferential circle (2 x π x radius) is right in the middle between two hexagons, and so increasing the sides of polygons sandwiching a circle both from outside and inside will also increase the digits of π. Truly amazing is that according to a historical book describing a calendar in the Sui dynasty compiled in 7th century in China, astronomer Zu Chong Zhi acquired approximately precise value for that time namely 3.1415926 < π < 3.1415927. In fact, the above book uses the same word as today, “circumference ratio”. In the beginning of 17th century, Rudolf van Keuren of Germany calculated the total length of sides of 32212254720 polygonal shape for over his entire lifetime and came to the correct value of π to 35 digits.This is the last great achievement in the time of polygon. In Japan Shigekiyo Muramatsu wrote the book “Sanso” and calculated π from the length of sides of polygon circumscribed about a circle: π nearly equals 3.1415 92648 77769 88692 48 to have calculated the correct value to 7 digits after decimal point. He calculated circumference ratio by his own way for the first time in Japan, independent from reference books imported from China.

• Thus, the time of calculating the ratio by simply increasing the sides of regular polygon came to end and entered new age of increasing the digits of π by improving calculating formula. Comprehensive achievement was brought by a mathematics genius of India, Srinivasa Ramanujan, (1887~1920), which was published as an “infinite series”. Fergason calculated the ratio to 540 digits by using the above formula in 1945. This was all done by his hand calculation.

• Now coming into computer’s age, the number was extended to the tremendous digits. With further improvement of computer, Yasumasa Kaneda calculated to amazing order of digits of 1,241.1 billion in 2002.

The other day when I visited, at Ikaruga Atelier in Tochigi prefecture, Mitsuo Ogawa-san, a carpenter who specializes in building shrines and temples, he kindly showed me the process of square wood becoming round pillar. It may be understandable that I suddenly came to think of π calculation while watching the piece of wood was transforming to octagon, hexadecimal polygon and thirty-two sides polygon.

I welcome your opinions on this column to the following E-mail address.
matogawa@planetary.or.jp