IV. Hexagonal Prototiles
Only edge-to-edge tilings are listed here. At this time, I'm aware of two distinct families of such tilings with hexagonal prototiles. One is an infinite family with several branches, and the other a pair of tilings based on the same prototile.
The tilings are classified as described in the Introduction. The figures available for each tiling are listed as links.
A. These tilings are a selection from an infinite family. These are grouped into a few different branches of the family. As a general rule, polygonal prototiles with more sides shrink more rapidly between successive generations. This results in less potential for overlapping of tiles, so that more tilings are allowed.
A1. This first branch is marked by v-shaped prototiles and fractal boundaries. In the first four, a single tile fits in the dip of the v.
- (p6, r6, g5, s.333, m2)
- (p6, r8, g5, s.293, m2)
- (p6, r8, g7, s.293, m2)
- (p6, r12, g11, s.268, m2)
In the next two, two tiles fit in the dip of the v.
- (p6, r5, g5, s.309, m1)
- (p6, r10, g9, s.201, m2)
A2. This branch is marked by v-shaped prototiles and polygonal boundaries.
- (p6, r3, g3/4, s.5, m1)
- (p6, r6, g3, s.5, m2)
A3. Another branch, closed related to the second quadrilateral family.
- (p6, r6, g9, s.289, m2)
A4. In this branch, the prototiles do not possess bilateral symmetry. The resulting tilings also do not possess any lines of mirror symmetry.
- (p6, r8, g9, s.207, m0)
- (p6, r12, g13, s.154, m0)
