Coloring booklets from finite subdivision rules

These coloring booklets come out of the research and programs of Bill Floyd, Jim Cannon, Ken Stephenson, and Walter Parry. Each booklet is based on a single finite subdivision rule, which is basically a combinatorial rule for subdividing a finite number of tiles, which are called tile types, into pieces which can be identified with tile types. One can then continue to subdivide further. The first figure of each booklet gives the subdivisions of the tile types, and the figures after that show further subdivisions of the tile types. The later figures are drawn using Ken Stephenson's program CirclePack.

The pentagonal subdivision rule
The single tile type is a pentagon. It's subdivision into six pentagons has rotational and reflectional symmetry. This is a beautiful example.

The twisted pentagonal subdivision rule
The single tile type is a pentagon. It's subdivision into five pentagons has rotational symmetry but not reflectional symmetry.

A twisted heptagonal subdivision rule
The single tile type is a heptagon. It's subdivision into eight heptagons has rotational symmetry but not reflectional symmetry.

The dodecahedral subdivision rule
This is a beautiful example with three tile types (a triangle, a quadrilateral, and a pentagon). The subdivisions of the tile types have rotational and reflectional symmetry. The coloring at the top of the page, which was computer drawn, is from this example.

A variant of the dodecahedral subdivision rule
This is a simplification of the dodecahedral subdivision rule.

Another variant of the dodecahedral subdivision rule
This is even simpler, and is inspired by the dodecahedral subdivision rule.

A rotationally symmetric subdivision rule with quadrilaterals and pentagons
There are two tiles types, a quadrilateral and a pentagon. The subdivsions of each have rotational symmetry but not dihedral symmetry.

The barycentric subdivision rule
The single tile type is a triangle, and it is subdivided into six triangles. As you keep subdividing, the valences of the vertices keep increasing.

The starburst subdivision rule
The single tile type is a hexagon, and it is subdivided into six hexagons. This example is closely related to the barycentric subdivision rule.

The diamond chains subdivision rule
There are two tile types, a quadrilateral and a triangle. The booklet gives subdivisions of the quadrilateral.

A hexagonal subdivision rule that isn't conformal
As you keep subdividing, a tile at one stage becomes a union of subtiles at later stages. Intuitively, a finite subdivision rule is conformal if the shapes of tiles don't get flattened as you keep subdividing. The pentagonal subdivision rule is a good example of a conformal finite subdivision rule.

Modifications of the Lattes subdivision rule
The Lattes subdivision rule has two tile types, A and B. A and B are each a square, and each is subdivided into four subsquares. Here are some examples obtained from the Lattes example by blowing up arcs.