One of my hobbies is playing around with different types of tilings. Mostly I like to play around with Penrose Kites and Darts. I also like to try to fit stars into regular patterns. The most recent exploration that I've done comes from Daud Sutton's "Islamic Design," where he talks about making a grid out of right triangles, and then placing regular polygons and other shapes onto the edges and corners of the triangles. In the case that I was interested in, he used five-pointed stars.
Getting the stars down on the triangles was simple enough, but it took me a lot of wiggling to get the nine kite-shapes at the top and bottom of the design to look symmetric and not smooshed.
I'll have to see what sorts of patterns will result from right-triangles which form squares instead of hexagons.
A couple of weeks ago, I read about a technique for putting odd-numbered polygons and stars together. Start with a figure, duplicate and reflect it, then make the two closest points touch. Skip a point on ether side of the touching point, and place a reflected duplicate there, too. This will make a repeating line, which you can put together into a mesh. I tried it with 5-, 7-, and 9-stars; the 7-stars were the most aesthetic, so I put together some interwoven 7-stars into a larger interwoven pattern.
Whats fun about this technique is that it allows one to break away from patterns that are hexagon- or square-based.
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