## TRIANGULATION

This blog is on a mathematical concept: Pascal’s triangle. This is an absurd thing for me to undertake as my knowledge of maths is based on an O level in 1952 with a touch of statistics in University, but now that I’m old I’ve realised it’s OK to be absurd and it’s absurd to try to be sensible all the time.

First the maths, which is very neat. Pascal built a triangle of numbers: at the top of the triangle is 1, the next line has 2 numbers 1 and 1, the following line 3 numbers 121 and so on

```1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1```

Add any two numbers on one line and the number below is the sum of those numbers. There are many more interesting things that can be done with this number pattern, take a look on https://www.mathsisfun.com/pascals-triangle.html .

Pascal’s triangle gets its name from the French mathematician and philosopher Blaise Pascal {19 June 1623 – 19 August 1662} but he wasn’t the first to discover it. In Iran the triangle is referred to as the Khayyam triangleas it was discovered by the Persian poet-astronomer-mathematician Omar Khayyam (1048–1131). He wasn’t alone though; Pascal’s triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian (1010–1070). In the 13th century, Yang Hui (1238–1298) presented the triangle and so it’s called Yang Hui’s triangle in China. In India it was known by Pingala, an ancient Indian poet and mathematician who lived around 300 BCE. He wrote the Chandaḥśāstra, where he analysed Sanskrit poetry mathematically and gave the first known explanations of Pascal’s triangle.

Pascal is also known as a philosopher and author of Pensees, a compilation of thoughts on life in general, but particularly on religion and the necessity of belief in Christ. He wrote the Pensees at a time of religious controversy in France when a sect known as the Jansenists, to which Pascal belonged, was at odds with the more powerful Jesuits. Omar Khayyam is best known as a poet who wrote the Rubaiyat (though it may not have been entirely his work). This long poem can be interpreted either as an example of mystical Sufism or of epicurean opposition to orthodoxy.

Many people today would feel mystified by the preoccupations of these men but imagine what Pascal, Khayyam, Yang Hui and Pingala would have felt if they could meet each other, how bemused each would have been by the others view on everything except the triangle that bears their names. It seems rather amazing to me that the properties of this triangle can be understood in any culture or century, and if our civilisation should fail and all knowledge was lost this triangle could be rediscovered anywhere at any time.

I am in awe of these mathematicians who were also poets and writers and their triangle full of possibilities. I offer a small poem.

Triangular Ode to Maths

```                              O
t o
s e e
d e e p
e a r t h
s t r i n g
b i n d i n g
g a l a x i e s
r e c o g n i s e
t e r r e s t i a l
i n t e r l a c i n g
m i r a c u l  o u s l y
t y i n g t h e  w o r l d
i n   m a t h e m a t i c a l
s k e i n s  o f  s u p r e m e
b e a u t y  a n d  b i n d i n g
w i t h  f a u l t l e s s  y a r n
t h e  d e c i p h e r e d   w o r l d  ```

Anne Bryan