I recently sat down and read through Structure and Interpretation of Computer Programs. Good book, need to go back and really digest it. While reading it, one of the examples made me as the question, “are all the points in a square the same distance from every other point in the square?” Turns out no, and I was able to prove it.

It took about half a page in my notebook and overall was just a fun bit of math. Very unlike the math I learned in high school, although it used many of the same principles. Early last year I read some of Barbara Oakley’s book, *A Mind for Numbers*. Instead of focusing on rote calculation, she instead teaches the relationship between numbers and concepts and emphasizes the benefits of both symbolic and geometric ways of describing a particular math concept.

I feel this approach makes much more sense. It feels both more approachable and more like what real mathematicians do.

That’s all I got on the topic, now here’s the proof:

Realize that any rectangle is just two right triangles smooshed up against each other.

Squares are rectangles with every side the same length. An equilateral rectangle, if you will.

The Pythagorean theorem states: A^{2} + B^{2} = C^{2}.

If the answer to the question “are all the points in a square the same distance from every other point in the square?” is to be yes,
then we would end up with N^{2} + N^{2} = N^{2}.

For some of you this is looking pretty solid already, but I’ll take it one step further to be really clear. Say X = N^{2}. We can then replace all the N^{2} with X and have an equivalent equation: X + X = X, or 2X = X. You can’t really have a square with sides of length 0 (the only number that statement is true for) as that’s actually just a point. So, no, the points in a square are not the same distance from every other point.

And that’s real math to me.