Pascal's Triangle

Pascal's Triangle is a very interesting pattern of numbers that was named after Blaise Pascal. Blaise Pascal was a french mathematician, physicist, inventor, writer, and a christian philosopher. He was educated by his father, and was a child prodigy.  He was one of the first inventors of the mechanical calculator which he called pascalines. He wrote a very influential treatise on the projective geometry when he was 16. Later in his life, he wrote significant works on philosophy and theology. His most famous works were Lettres provinciales and Pensees. Pascal died at the age of 39 two months after his birthday.

 Building the triangle
1. Place a 1 at the top of the triangle.
2. Continue to add more rows in a triangular pattern. Each new number should be the sum of the two numbers right above it.
For ex. In the fourth row of Pascal's triangle on the far left 1 and 3 are right next to each other. The number right below the two will be 4 since 1+3=4.


3. Continue the pattern

Patterns in the triangle

1. Diagonal Patterns


a. The first diagonal just contains ones. (1,1,1,1,1,etc...)
b. The second diagonal is counting numbers starting at 1 and continuing on. (1,2,3,4,5,etc...)
c. The third diagonal has triangular numbers starting at 1 and continuing on. (1,3,6,10,15,etc...) Triangular numbers  are formed by the number of dots in a triangular pattern.






d.The fourth diagonal has tetrahedral numbers starting at 1 and continuing on. (1,4,10,20,35,etc...) Tetrahedral numbers are formed by the number of dots in a tetrahedron pattern.

2. Symmetrical Pattern

Pascal's triangle is symmetrical. The numbers on both sides mirror each other. For example, in the third row of Pascal's triangle the numbers are 1,3,3,1. The numbers on each side are identical.
 

3. Horizontal Sums











The sum of each horizontal line doubles every time. They are powers of two.
(1, 2, 4, 8, 16, 32, 64, 128)

4. Exponents of 11









Each horizontal line in Pascal's triangle is an exponent of 11.
a. First line: 11⁰ = 1
b. Second line:11¹ = 11
c. Third line: 11² = 121
d. Fourth line: 11³ =13331
e. Fifth line: 11⁴ = 14641
f. Sixth line: 11⁵ = 161051
The sixth line is a little different. In Pascal's triangle in the sixth horizontal lines the numbers are 15101051. Certain digits overlap and are added together. 

5.Squares

In the second diagonal of Pascal's triangle, the number next to and below a number in the diagonal is equal to that numbers square.
a. 3² = 3 + 6 = 9
b. 4² = 6 + 10 =  16 

c. 5² = 10 + 15 = 25



6. Fibonacci Sequence

If you add a number in the first diagonal of Pascal's triangle with the numbers if you go up and across 1 box, you will have fibonacci numbers. For example, all the numbers in the dark blue which are 1, 6, 10, and 4 you get 21 which is a fibonacci number.  

7. Odds and Evens


Color the odd and even numbers in Pascal's triangle different colors. It will create a new pattern matching the Sierpinski triangle.

My triangle.