You mock, but this kind of tinkering is basically the foundation of mathematics.

I suggest reading Lockhart's Lament, linked on this Wikipedia page: https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament

Relevant quote:

> For example, if I’m in the mood to think about shapes— and I often am— I might imagine a triangle inside a rectangular box. I wonder how much of the box the triangle takes up? Two-thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. There’s no ulterior practical purpose here. I’m just playing. That’s what math is— wondering, playing, amusing yourself with your imagination.