The Math Less Travelled, one of the blogs that I read, posted yesterday a link to another web site, a Tiling Database created by Brian Wichmann and Tony Lee. The database is exactly what it says on the label: a collection of patterns which one could put on a flat surface and extend outward in both directions as far as you like. In principle, you could get any of them to spruce up your kitchen, although some of them would be a bit staggering to face in the morning, even in other color schemes.
You can pull up a display of twenty randomly selected patterns easily, but what got me hypnotized for a while was the Simple Tile Search. That search asks, simply, what sorts of regular polygons — polygons where each side has the same length — and what regular star-shaped polygons — polygons where each side has the same length, and it looks like a star — there should be in the tile: triangles, squares, four-pointed stars, nine-pointed stars, whatnot. You can put together bundles of them, so, search for patterns that have triangles, hexagons, and ten-sided stars, say. There are none. Well, let’s say triangles, hexagons, and four-pointed stars. That pulls up two patterns, both given descriptions that say where they come from — quite a few have artistic origins — and you can get descriptions of the geometry involved.
There’s a lot of interesting things in mathematics regarding symmetry, where symmetry can be the things we normally think of, such as mirror symmetries, or rotational symmetries (like when you turn a crossword puzzle around and the pattern of black and white squares turns out to be the same), and some that take some time to get used to, particularly, a translational symmetry (if you pick up the pattern and move it a fixed distance to the side and up-or-down, does it look the same, which is of key importance in tiling the kitchen this way). The descriptions of the patterns turns up lists of symmetry groups, which describe what those symmetries in the pattern are.
Never mind them. What got me going was the simple puzzle: what collections of desired polygons and stars turn up tiles? What collections don’t? I can’t have triangles, hexagons, and ten-pointed stars; how about triangles, hexagons, and eight-pointed stars? Six-pointed stars? Can I have triangles, squares, and pentagons in the same tile? How about triangles, squares, and hexagons?
This is play, certainly. It’s also where the mind starts to form hypotheses: supposing that there rules about what groupings can possibly go together and what ones can’t, can we discern any of them? It wouldn’t be a satisfactory proof to say that since according to this search of tile patterns — drawn from what references Wichmann and Lee were able to gather, and what they’ve had sent to them — there’s no pattern with triangles, squares, and octagons then they can’t possibly go together. But with the suspicion that it is impossible, one could go into the harder details of symmetry groups and their connected fields and come up with a logically rigorous proof that such a pattern can’t exist, or perhaps show how it can, and make a new entry for The database.
This benefit — I first wrote this tangible benefit, but proving or disproving the existence of a particular tiling pattern probably isn’t all that tangible — is a little extra reward that we might reach at the end of some study. The hypnotic playing, the forming of guesses and the rapid confirming or extinguishing of them as an obvious counterexample comes up, is the start of study and part of what makes mathematics fun.