Mathematical Triangle

There are two famous types of triangles in Mathematics and Number theory. The first one is the Pascal’s Triangle. Pascal's triangle is a symmetric sequence of numbers arranged, well, like a triangle. First, let’s go over how the triangle is found.

This triangle is found by taking the sum of the numbers above it. Like most sequences, it starts with the number 1. The next row will just have 1’s because 1 + nothing = 1. The next row is generated the same way. The 2 is found by adding the ones. As you can see, the outer diagonal sides will always be one’s because there is no other number to add to it. There are a couple of cool patterns and sequences that can then be generated from this triangle. The first has to do with the diagonals. The first diagonal is all ones, the next is all of the positive whole numbers in ascending order. The next diagonal is a little bit trickier, but it is all of the Triangular numbers. Basically, it is the number of dots needed to make a right triangle if you increase the height of the triangle by one each time.

One more cool feature of this triangle is that coloring all of the odd and even numbers different colors will result in Sierpinski's triangle which has other applications and features. But what is Sierpinski's triangle?

This is! It is another common fractal or a repeating shape. It is found by starting with a triangle and adding the same-sized triangles on each corner of that. Each above shape is duplicated and added to the one below it. There are a ton of cool graphics that continually zoom in to this shape, and it will continue to look the same. Because this is a self-similar fractal, it can also be created by finding the midpoints of each of the sides of the largest triangles and connecting them. What is interesting is that this same pattern can apply to shapes other than the triangle. For example, the thumbnail for this post is a duck with other ducks in. You start by drawing your original shape and then drawing the next biggest duck inside the original shape. Continuing with that pattern, you get a repeating duck fractal. The one difference is that the triangle is self-similar and symmetric because of its equilateral nature.

Now why, might you ask, does this fractal have any use. Well, there are a couple of reasons. The first is that it demonstrates an exponentially increasing function. Each time a new triangle is added, it takes all of the previous ones + some number. Another application of this triangle is that it represents other increasing puzzles. One such puzzle deals with stacking rings on poles. Each ring added increases the number of moves to move that stack of rings to another pole. When mapped out, this very closely resembles the Sierpinski’s Triangle. One real-life application is that this fractal is used in antenna designs. A fractal design has the advantages of multiple “resonances or enhancing bandwidth and hence it can be used in cognitive radio for spectrum sensing.” My guess is that the self-similar nature optimized the way that signals are read.

Fractals like these and sequences Pascal’s triangle are useful for developing relationships between numbers and optimizing certain features in the developing world.