Pascal's Triangle
This post is about a very interesting number pattern in mathematics known as Pascal’s Triangle, named after the French Mathematician Blaise Pascal.
Pascal’s Triangle is a peculiar number pattern in mathematics which is formed by starting with 1 on the top row.
To form the triangle, start with a 1 at the top. Then, you continue placing the rest of the numbers in a triangular fashion such that each number is equal to the sum of the 2 numbers directly above it.
There are a lot of interesting patterns hidden within the Pascal’s Triangle, some of which I will explain in this post.
Symmetry
The first and obvious thing that will strike you is the symmetry of the Pascal’s Triangle
Powers of 11
Another peculiar aspect you will notice is that, each row of the Pascal’s Triangle represents the powers of 11. Yes
The 1st row represents 11^0 = 1
2nd row represents 11^1 = 11
3rd row represents 11^2 = 121 and so on
If you see, the 6th row has 1, 5, 10, 10, 5, 1. But 11^5 = 161051. They don’t match, you might think. But they do match. The only thing is that the 5 & 10 get combined into 6105, thereby giving 161051
The Binomial Expression
Another interesting thing to notice is that the terms in any row of the Pascal Triangle correspond to the co-efficient’s of the binomial expansion
All of us know the famous formula for expansion of sum of 2 cubes:
What are the co-efficients? 1, 2 & 1. This corresponds with the 3rd row of the Pascal’s Triangle. Interesting right?
Consider the 4th row: 1, 3, 3, 1. What is the expansion of (a + b)3? There you have the answer!!!
Powers of 2
Sum the numbers in any row of the Pascal’s Triangle and you will find that they are always powers of 2 (in increasing order as you go down)
The reason as to why this occurs, closely ties up with the binomial expansion. Substitute a=b=1 in the binomial formula in the previous table and you would get the answer.
Sierpinski Triangle
The Sierpinski Triangle is a fractal described by Waclaw Sierpinski in 1915. It is a self similar structure that occurs at different levels of iterations, or magnifications.
Take an equilateral triangle and join the mid points of the sides. This will divide the triangle into 4 parts of equal area. Now repeat the same in each of the 4 smaller triangles. The resultant figure you get is called a Sierpenski Triangle
Now, take the Pascal Triangle. Colour only the odd numbers. You will find that the resultant figure you get will be very similar to the Sierpinski Triangle.
Singmaster’s Conjecture
Another interesting fact is that, there is an unsolved problem in mathematics concerning the Pascal’s Triangle called the Singmaster’s Conjecture named after the British mathematician David Singmaster who proposed it in 1971.
Simply put, it states that there is a finite upper bound on the number of times a number occurs in the Pascal’s Triangle (except 1, which occurs infinite times)
These are some of the observations that can be made from the Pascal’s Triangle. If you know any other such interesting observations, do drop them in the comments.
Thanks for reading!!!