Natalie Priebe Frank
Some of my favorite tiling
substitutions
Some of the famous examples
The Chair tiling is made from four rotations of the same tile. You can consider the rotations distinct or not, depending on your purposes.
Here are some iterations of the Chair substitution:
The Penrose Tiling comes in numerous flavors. Here's one version, with marked rhombi:
If you try and iterate this, you'll notice that some of the tiles overlap. Don't worry about it: since the overlapping tiles are identical, consider them just one tile. Here are a few iterations of a star of five tiles.
The Pinwheel tiling is an example where the tiles appear in an infinite number of rotations. Here is the substitution rule, followed b a level-3 supertile:
Not--so--famous examples
A tiling with three possible fault lines
The substitution for the background of the page (discovered by N.P.F.). Notice that the substitutions are not geometrically similar to the tiles they are replacing. Still, one can iterate the substitution.
A few iterations of the horizontal blue rectangle:
An interesting thing about this example is that you can use a limit process to get tiles that are geometrically similar to their replacements. You basically just change the side lengths. Here is what a few iterations of the blue rectangle look like in that case:
Compare and contrast the images. They have the same number of each tile type in approximately the same locations, but the adjacencies are all different. If you know what a fault line is, this tiling allows them in three different directions.
A tiling that fakes randomness in a technical way (spectral-theoretically)
If you iterate this next substitution, you get a substitution with absolutely continuous dynamical spectrum (another one discovered by me):
Identifying the greens together and the reds to white is what makes the continuous part:
