Recently he posted this:
I misunderstood it. I thought it was a single triangular tile, sometimes flipped and sometimes rotated. So I got playing around with it, trying to generalize to other cases than the hexagonal. I made a simple triangle, with two triangles in it. The red has vertices on the border, and the blue has vertices in the region. (The commands are Point[<object>] for the first, and PointIn[<region>] for the second.) The other command is Sequence. The basic structure is usually something like Sequence[Rotate[thing, i*360o/n, centerpt],i,1,n].
GeoGebraGeek paragraph: The fine point in this one is I wanted it do do something differently for 2, 4, 6 than for 8, 10, 12. So I defined a Boolean variable, a = n >7. Booleans read as both true/false and 1/0. The numeric is handy for dynamic coloring (eg. turn something green if it's where you want it) or for goofy stuff like this; Sequence[Rotate[thing, i*(1+a)*360o/n, centerpt],i,1,n/(1+a)]. That makes it skip every other side for the values where they'd overlap.
Then some motions and - voilà - you get a kind of tiling.
With World Tessellation Day coming up, I've been thinking about tessellations a lot. I've mostly concentrated on Escher style, but I think I want to look more into decorating plainer tiles and then seeing the results of the symmetries, too.
The sketch is on tube.geogebra.org for play, of course. (Some of my other tessellations are in this geogebrabook.)
Now I'm thinking one of the decorations should have been a quadrilateral. Next time!
A gif of using the sketch; A decagonal symmetry.
Square symmetry and dodecagonal for the same decorations.