In the middle of playing around with various tiles and patterns, I lucked onto a post from Alex Bellos from back in February, in which he gives instructions for some decorative geometric constructions. He suggests dusting off a geometry set to implement them, but I found GSP worked nicely.
If you start with the line AB and follow Alex's instructions, you'll end up something like the above. Hiding what we want to hide, and mapping A and B onto the other corners of the square using an iteration will give you something that looks very close to his finished product.
In more decorative versions, the edges that extend out weave together in a braided fashion, producing an even more striking effect. But it was one of the later examples from the post that I was more interested in, as it seemed to show a more general method of using overlapping regular polygons (in this case, regular dodecagons) to produce decorative patterns. In one example, the dodecagons intersect at the midpoints of their sides at 90 degree angles:
I found that having the dodecagons intersect at the midpoints of their sides at 30 and 60 degree angles produces another interesting pattern.
If instead of intersecting at midpoints, we intersect the dodecagons at vertices, we can get other patterns. Here's one where the dodecagons have the smallest of overlaps.
Overlapping a little more, you get a pattern that you can make from a standard set of pattern blocks.
Still with a greater overlap, you can make a nice rosette.
And this pattern below has dodecagons overlapping in two ways (a bit hard to see the second):
Well, that was fun. Any other decorative patterns from overlapping dodecagons? I'll finish off with that dodecagon rosette pattern around a dodecagon.