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.
In this tessellation, regular polygons have been given the brighter colors, while the two non-regular polygons have pastel colors.
nebusresearch 
Reminiscent of Maurice Escher’s drawings!
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Ooh, now, it is, isn’t it? I didn’t think of that.
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I really like your abstract algebra explanation. (I didn’t like the one semester of abstract algebra I took in college, but now I’m thinking I judged it too quickly. You make it sound fascinating!)
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Thank you kindly. Abstract algebra, I have to admit, I didn’t really like my first semester (although I had a great professor, one who was really encouraging and supportive and who had, honest to goodness, a twinkle in his eye when talking about the subject), but I did warm up to it.
I suspect it’s hard getting into because, at least as I remember, it’s kind of the first course where there aren’t really practice problems that can be done. I mean, when you learn calculus, you can practice differentiating x2 and x2 + x and x3 and so on, checking back with someone who knows the answers, until you’re really good at differentiating. Come abstract algebra, though, and you get homeworks that are mostly proofs, well, you either show this divisibility theorem is true by a sound argument or you don’t, and there’s not a variation you can do to practice the skills you need.
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