Uniform Tiling - Expanded Lists of Uniform Tilings

Expanded Lists of Uniform Tilings

There are a number ways the list of uniform tilings can be expanded:

1. Vertex figures can have retrograde faces and turn around the vertex more than once.
2. Star polygons tiles can be included.
3. Apeirogons, {∞}, can be used as tiling faces.
4. The restriction that tiles meet edge-to-edge can be relaxed, allowing additional tilings such as the Pythagorean tiling.

Symmetry group triangles with retrogrades include:

(4/3 4/3 2) (6 3/2 2) (6/5 3 2) (6 6/5 3) (6 6 3/2)

Symmetry group triangles with infinity include:

(4 4/3 ∞) (3/2 3 ∞) (6 6/5 ∞) (3 3/2 ∞)

Branko Grünbaum, in the 1987 book Tilings and patterns, in section 12.3 enumerates a list of 25 uniform tilings, including the 11 convex forms, and adds 14 more he calls hollow tilings which included the first two expansions above, star polygon faces and vertex figures.

H.S.M. Coxeter et al., in the 1954 paper 'Uniform polyhedra', in Table 8: Uniform Tessellations, uses the first three expansions and enumerates a total of 38 uniform tilings.

Finally, if a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.

The 7 new tilings with {∞} tiles, given by vertex figure and Wythoff symbol are:

1. ∞.∞ (Two half-plane tiles, infinite dihedron)
2. 4.4.∞ - ∞ 2 | 2 (Apeirogonal prism)
3. 3.3.3.∞ - | 2 2 ∞ (Apeirogonal antiprism)
4. 4.∞.4/3.∞ - 4/3 4 | ∞ (alternate square tiling)
5. 3.∞.3.∞.3.∞ - 3/2 | 3 ∞ (alternate triangular tiling)
6. 6.∞.6/5.∞ - 6/5 6 | ∞ (alternate trihexagonal tiling with only hexagons)
7. ∞.3.∞.3/2 - 3/2 3 | ∞ (alternate trihexagonal tiling with only triangles)

The remaining list includes 21 tilings, 7 with {∞} tiles (apeirogons). Drawn as edge-graphs there are only 14 unique tilings, and the first is identical to the 3.4.6.4 tiling.

The 21 grouped by shared edge graphs, given by vertex figures and Wythoff symbol, are:

1. Type 1
   • 3/2.12.6.12 - 3/2 6 | 6
   • 4.12.4/3.12/11 - 2 6 (3/2 3) |
2. Type 2
   • 8/3.4.8/3.∞ - 4 ∞ | 4/3
   • 8/3.8.8/5.8/7 - 4/3 4 (2 ∞) |
   • 8.4/3.8.∞ - 4/3 ∞ | 4
3. Type 3
   • 12/5.6.12/5.∞ - 6 ∞ | 6/5
   • 12/5.12.12/7.12/11 - 6/5 6 (3 ∞) |
   • 12.6/5.12.∞ - 6/5 ∞ | 6
4. Type 4
   • 12/5.3.12/5.6/5 - 3 6 | 6/5
   • 12/5.4.12/7.4/3 - 2 6/5 (3/2 3) |
   • 4.3/2.4.6/5 - 3/2 6 | 2
5. Type 5
   • 8.8/3.∞ - 4/3 4 ∞ |
6. Type 6
   • 12.12/5.∞ - 6/5 6 ∞ |
7. Type 7
   • 8.4/3.8/5 - 2 4/3 4 |
8. Type 8
   • 6.4/3.12/7 - 2 3 6/5 |
9. Type 9
   • 12.6/5.12/7 - 3 6/5 6 |
10. Type 10
    • 4.8/5.8/5 - 2 4 | 4/3
11. Type 11
    • 12/5.12/5.3/2 - 2 3 | 6/5
12. Type 12
    • 4.4.3/2.3/2.3/2 - non-Wythoffian
13. Type 13
    • 4.3/2.4.3/2.3/2 - | 2 4/3 4/3 (snub)
14. Type 14
    • 3.4.3.4/3.3.∞ - | 4/3 4 ∞ (snub)