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# Pascal's and Sierpinski's Triangles and Their Relationship

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## Seixas Aldrich

on 4 March 2015

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#### Transcript of Pascal's and Sierpinski's Triangles and Their Relationship

Patterns in Pascal's Triangle #1
The first Pascal Triangle pattern I am doing is the "1s" pattern. In this the outside number is always the number 1.
Interesting Discovery
The chinese discovered the Pascals triangle before anyone else, as shown in the picture that is part of Chu Shi-Chieh's book "Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements) that was written way before anyone in other countries found out about the triangle. Even before Blaise Pascal was even born.
What is the Relationship Between Pascal's Triangle and Sierpinski's Triangle
The Relationship between the two triangles are that if you shade in all the odd numbers in Pascal's Triangle in one color and leave the even numbers in another color it makes Sierpinski's Triangle.
Pascal's Triangle
Sierpinski's Triangle
Sierpinski's Triangle is a set of triangles named after the mathematician Waclaw Sierpinski. To make this triangle you first start out with an equilateral triangle. Then you put another equilateral triangle with the base on top and each corner bisecting each side of the first triangle. Repeat with the three triangles made from the second one, and so forth.
Pascal's and Sierpinski's Triangles and Their Relationships
Patterns in Pascal's Triangle #2
This pattern is called squares. In Pascal's Triangle, in the second diagonal, the square of the number is the number to the right of it and under it to the right added together.
Where does the Fibonacci exist in Pascal's Triangle
The Fibonacci Sequence is found in Pascal's Triangle when you add up the diagonals (like in the picture below), it makes the Fibonacci Sequence.
By: Seixas Aldrich
Patterns in Pascal's Triangle #3
The next one is called triangle numbers. This one is the is the third diagonal row, it goes 1, 3, 6, 10, 15, 21... It is called triangle numbers because the numbers is from a sequence of dots that make triangles.
Patterns in Pascal's Triangle #4
This one is called the hockey stick pattern. In Pascal's Triangle, if a diagonal of numbers of any length is selected starting at any of the 1's on the sides and ending on any number inside the triangle on that diagonal, the numbers inside the diagonal added up is equal to the number below the end of the diagonal that is not on the same diagonal itself.
Patterns in Pascal's Triangle #5
Prime numbers is this ones name. This pattern is whenever the second number in a row is a prime number, all of the other numbers have to be a divisible by it(other than "1").
Patterns in Pascal's Triangle #6
This next pattern is called powers of 2. The first row is 1, which is 2 to the 0th power. The next row is 1 and 1 which adds up to 2 which is 2 to the 1st power. It goes on with you adding the row and it is the powers of two in order.
Patterns in Pascal's Triangle #7
The next pattern is the plus minus pattern. In this pattern you take any row and put a plus sign in between the first two numbers. Then you put a minus sign in between the next tow numbers and so on. When you do out that expression it always ends up as 2 (except the first row).
Patterns in Pascal's Triangle #8
This pattern in contrast to the last on is the minus plus pattern. This pattern is similar to the last one but it goes minus then plus after every number and not plus than minus s in the picture below. Also at the end it always ends up at 0.
Patterns in Pascal's Triangle #9
This one is called magic 11. This is similar to the powers of two. The first row is 11 to the 0th power, 1. The to the 1st, 11. Second, 121. It goes on until it reaches row six where it goes 1 5 10 10 5 1, and 11 to the 5th power is 161,051. Since 10 has two digits you have to carry over to the left the 1 and then the 1 from the second 10, and it gets to 161,051
Patterns in Pascal's Triangle #10
The last pattern is called heads and tails pattern. In this pattern you can find the probability of flipping a coin a certain amount of times. You start out with the second row with flipping the coin one time, it lands on heads once and tails once, so 1 1. Then row two with two flips and it can land head head, head tail, tail head, or tail tail. This is 1 2 1. It goes on with Pascal's Triangle.
Pascal's Triangle is a triangle composed of numbers in a specific order named after Blaise Pascal. To build the triangle you have to start with the number "1" at the top then place numbers under it in a triangle. The numbers under are the two numbers above it added together.
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