M.C. Escher is a great example of math in art; though he had no head for the details of math, he had a great visual intuition for a number of mathematical concepts, and he loved to use his art in order to represent them in a clean and simple manner. He had a special love for tessellations: a shape or set of shapes that can be tiled infinitely to cover a plane. This tessellation of his in particular — Circle Limit III — demonstrates very simply an example of non-Euclidean geometry; specifically, he drew this tessellation as if it was in the hyperbolic plane rather than the traditional one. Notice how they get more scrunched together as they get closer to the edge, this represents that the hyperbolic plane contains infinite area within a finite bound. Here, the swooping white lines down the centers of the tiles are (essentially, though not quite) the equivalent of straight lines in the hyperbolic plane.
Hyperbolic geometry is one of two major non-Euclidean forms of geometry. (There’s Euclid again!) In normal Euclidean geometry, if you have a line and a point, there’s exactly one new line through that point parallel to the first. This is also called “flat curvature”. In spherical geometry, there are no parallel lines through that point; we also call that “positive curvature”. While here, in hyperbolic geometry, there are an infinite number of parallel lines through the point, and we call this “negative curvature”.
We can also notice an interesting fact about triangles in the hyperbolic plane through this art. Of course, in the normal plane, all triangles without question have a total interior angle sum of 180°. Here, though, triangles have a total interior angle sum of ≤180°; closer to the center, it’s closer to “straight”, while closer to the edges the triangles are more skewed. Similarly, in the spherical plane, triangles have a total interior angle sum of ≥180°
This all ties into Euclid’s Parallel Postulate, which I’ll cover in a later post, along with covering both these variants of geometry in more detail.