By Evierose Grace
Mathematical patterns are unquestionably thought-provoking. One that I find particularly fascinating is Pascal's Triangle.
Pascal’s triangle is a simple triangular pattern, commencing with a ‘1’ at the top. Then each of the following cells are the sum of the two cells above.
History of Pascal’s Triangle
‘Pascal’s triangle’ is named after Blaise Pascal, a French mathematician. He was one of the initial investigators of this pattern, in Europe. However, it had been understood in other cultures like India, Greece, China, and Iran, for centuries before. The pattern studied in ancient Chinese culture is very similar to the one we have today.
Blaise Pascal first began studying this tessellation in 1654. When Pascal was constituting his own triangle he based it off of how Chu Shi-Kie formulated his in the 1300’s. Pascal’s original triangle was slightly different to the one that is most common today, and being drafted at a different angle, it also had an extra set of diagonal lines on the exterior, mimicking the third diagonal set.
The Hidden Triangles in Pascal’s Triangle
Mathematically Pascal’s Triangle is captivating because within the unabridged, outermost triangle, there are numerous smaller triangles. For example:
~If you highlighted all of the numbers divisible by a ’2’, ‘3’, ‘4’, or ‘5’ you would find that the multiples fashion smaller triangles.
As well as containing smaller triangles, this intriguing diagram houses a multitude of riveting patterns, such as:
-The cells in the third diagonal lines on both sides are the triangle numbers.
-The cells in the fourth diagonal lines on both sides are the tetrahedral numbers.
-The sum of all the numbers in a row are the powers of ‘2.’
-In the rows that contain a prime number in the second cell, all
numbers in that row (excluding the ‘1’) are multiples of that prime number.
You can also find the Fibonacci sequence inside it, by adding up each left to right diagonal line. Making Pascal’s Triangle a useful tool for remembering the Fibonacci sequence, and vice versa!
No wonder the human race has spent thousands of years studying this pattern, it’s fascinating!