For my first series of blog posts, I thought it would be fun to introduce my favorite piece of mathematics.
Ok, so, it’s not actually my favorite piece of mathematics (how could I pick just one?), but it is my favorite piece of mathematics to explain, and it’s a great excuse to flex the muscles on this static site blog template I found.
I believe the mathematics below is incredibly fun to show people because:
It deals with geometry.
Geometry is the class in school that is often the most thoroughly-wrung, having all actual mathematics removed and replaced with a dry, foul tasting powder that is most toxic to childhood curiosity.
It plays with definition.
Many people come to believe that mathematics is about memorizing “the one true set” of definitions. If you’ve ever seen those viral faux-math problems that incite facebook comments to endless debate about order of operations, you will have seen this at play. Doing mathematics (with friends, coworkers, in a research paper, or on your own) is about choosing the definitions you want to work with, and seeing where those definitions take you.
It is surprising!
At the end of this 3 part series, we’ll reach a conclusion that has absolutely no right being true, but unavoidably is. We will have made a creative choice in how we define things, yet still we will be lead inexoribly to this conclusion.
I first saw the mathematics I’m about to show you, presented in a similar way, in a wonderful book called The Art of the Infinite. If you don’t normally enjoy math, but you enjoy this series of blog posts, I cannot recommend this book enough.
This piece of mathematics involves triangles:
However, I don’t want to just talk about just this triangle, I want to talk about any triangle.
Feel free to drag the points above and pick your own triangle!
The piece of mathematics I want to show you today involves the center of this triangle. But what does “center” mean here?
One of my favorite things about mathematics is that it is a playground of definitions: to do mathematics is to choose a set of definitions and follow the roads those choices build for you.
So what are some ways we could define the center? One reasonable to define the “center” of a triangle is to somehow combine the centers of it’s edges:
One way we might combine these midpoints is by extending a perpendicular line from them. I’ll start with just one:
There is something powerful under the surface here. These diagrams are merely representations of the toys we are actually playing with: grainy photos of the infinitely thin lines and infinitely precise points.
We said this new perpendicular line intersects the edge of our triangle exactly half-way; if you count the individual pixels of the graphic, you may find that to be false, but the object in both of our minds right now is unflawed and perfect.
Under a particularly philosophical moon, you might find me pointing out that mathematics is, in some deep fundamental way, the only time we are allowed to flirt directly with perfection.
But, tonight is just a regular moon, so lets explore the consequences that fell out of our choice. The first one is that any point on this perpendicular line must be an equal distance from either of the two points at the ends of the line it bisects.
Feel free to drag the various points around, and convince yourself of why that would be the case.
Let’s add the perpendicular bisectors for the other sides of the triangle, shown below in purple and orange:
This is a good place for me to end the first part of this series of blog posts with a question.
Before moving on to part two, I encourage you to ponder the little triangle near the intersection of these three perpendicular bisectors: What can we conclude about that triangle, without a shadow of a doubt?