I’m afraid I lack the time to talk about this in more detail today, but, Robert Loves Pi, a geometry-oriented blog, has a lovely tessellation that you might like to see. Tessellations are ways to cover a surface, usually a plane, with an, ideally, small set of a couple pieces infinitely repeated. As a field of mathematics it’s more closely related to kitchen floors than the usual, but it’s also wonderfully artistic, and the study of these patterns brings one into abstract algebra.
In abstract algebra you look at things that work, in some ways, like arithmetic does — you can add and multiply things — without necessarily being arithmetic. The things that you can do to a pattern without changing it — sliding it in some direction, rotating it some angle, maybe reflecting it across some dividing line — can often be added together and multiplied in ways that look strikingly like what you do with regular old numbers, which is part of why this is a field that’s fascinating both when you first look at it and when you get deeply into its study.