Monday, February 8, 2016

Illuminations 2.8.16


One of my favorite things in this world is math. Now, hush your boos and put those tomatoes back in the fridge. I see you. Let me explain myself.

Math, I’d like to say, is the universal truth. We all know and accept that one thing and another thing together make two things, and two things and another two things make four things. That’s basic addition and one of the most axiomatic things that you don’t have to prove. Every other application in mathematics stems from this truth; multiplication is just repetitive addition, subtraction is just about the reverse of addition, and even complex operations like integrals and derivatives come from a basic understanding of putting two and two together. Also, do you see all these texts and colors on your computer screen? That's just your computer doing hardcore addition.

I’m not here to give you a math lecture, and honestly I’m not at all qualified to do so, but I am here to help you appreciate math in at least one form: it’s aesthetic qualities.

We’re all familiar with the the Cartesian Graph, right? The classic rectangular coordinate grid where a point is defined by its ‘x’ and ‘y’ location; formulas would be in the form of: y = mx +b. Well there’s another system that creates beautiful graphs with simple formulas; It’s called the Polar Coordinate System and it’s used to map out distances from a central point as an angle increases. Functions in this system often create amazing symmetrical graphs. Here are a few:



This one's actually a comparison of a Polar Graph to a Cartesian one with the same formula.
You have no idea how much time I’ve spent in my high school calculus class playing around with a graphing calculator and trying to figure out how to make some of these curves.

Something else you might be familiar with is the numerical pattern, Pascal’s Triangle. For those who don’t know what this is, it’s a triangle of numbers where each number is a sum of the two numbers directly above it.


You might be thinking, “so what? It’s a group of numbers in the shape of a triangle,” but Pascal's Triangle is much more than that. It’s usefulness in mathematics is very deceiving. Besides being a great introduction into numerical sequences, Pascal’s Triangle has its usefulness in calculus for creating binomial functions and in discrete mathematics and probability for it’s graphical representation of combinations. One of my favorite things about Pascal’s Triangle is that if you differentiate the even and the odd numbers in the sequence by coloring in the odd numbers and leaving the even numbers alone you’re left with Sierpinski's Triangle, a fractal shape that is created by drawing small triangles within newly created triangles.



Math is an amazing and useful thing, but for our sake let’s just call it a beautiful thing.





-Christian

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