List of uniform tilings

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This table shows the 11 convex uniform tilings of the Euclidean plane, and their dual tilings.

There are three regular, and eight semiregular, tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face.

Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex.

These 11 uniform tilings have 32 different uniform colorings. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are NOT color uniform!)

In addition to the 11 convex uniform tilings, there are also 14 nonconvex forms, using star polygons, and reverse orientation vertex configurations.

Dual tilings are listed by their face configuration, the number of faces at each vertex of a face. For example V4.8.8 means isosceles triangle tiles with one corner with 4 triangles, and two corners containing 8 triangles.

In the 1987 book, Tilings and patterns, Branko Grünbaum calls the vertex uniform tilings as Archimedean in parallel to the Archimedean solids, and the dual tilings Laves tilings in honor of crystalographer Fritz Laves.

Contents 1 Convex uniform tilings of the Euclidean plane 1.1 The R3 [4,4] group family 1.2 The V3 [6,3] group family 1.3 Non-Wythoffian uniform tiling 2 Expanded lists of uniform tilings 3 Uniform tilings in hyperbolic plane 3.1 The [7,3] group family 3.2 The [5,4] group family 3.3 (4 3 3) family 4 See also 5 External links 6 References

[edit] Convex uniform tilings of the Euclidean plane

[edit] The R₃ [4,4] group family

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings
Square tiling	4.4.4 4 | 2 4 p4m	self-dual
Truncated square tiling	4.8.8 2 | 4 4 | 4 4 2 p4m	Tetrakis square tiling
Snub square tiling	3.3.4.3.4 | 4 4 2 p4g	Cairo pentagonal tiling

[edit] The V₃ [6,3] group family

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings
Hexagonal tiling	6.6.6 3 | 6 2 2 6 | 3 3 3 3 | p6m	Triangular tiling
Trihexagonal tiling	3.6.3.6 2 | 6 3 3 3 | 3 p6m	Quasiregular rhombic tiling
Truncated hexagonal tiling	3.14.14 2 3 | 6 3 3 | 3 p6m	Triakis triangular tiling
Triangular tiling	3.3.3.3.3.3 6 | 3 2 3 | 3 3 | 3 3 3 p6m	Hexagonal tiling
Small rhombitrihexagonal tiling	3.4.6.4 3 | 6 2 p6m	Deltoidal trihexagonal tiling
Great rhombitrihexagonal tiling	4.6.12 2 6 3 | p6m	Bisected hexagonal tiling
Snub hexagonal tiling	3.3.3.3.6 | 6 3 2 p6	Floret pentagonal tiling

[edit] Non-Wythoffian uniform tiling

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings
Elongated triangular tiling	3.3.3.4.4 2 | 2 (2 2) cmm	Prismatic pentagonal tiling

[edit] 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.

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 all of these 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 (an infinite prism)
3. 3.3.3.∞.∞ | 2 2 ∞ (an infinite 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. 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
   • 8/3.8.8/9.8/7 - 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 - snub
14. Type 14
    • 3.4.3.4/3.3.∞ - snub

[edit] Uniform tilings in hyperbolic plane

There are an infinite number of uniform tilings on the hyperbolic plane based on the (p q 2) hyperbolic regular tilings.

Shown with Poincaré disk model are two families:

[edit] The [7,3] group family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual tilings
Order-3 heptagonal tiling	7.7.7 3 | 7 2 [7,3]	Order-7 triangular tiling
Order-3 truncated heptagonal tiling	3.14.14 2 3 | 7 [7,3]	Order-7 triakis triangular tiling
Triheptagonal tiling	3.7.3.7 2 | 7 3 [7,3]	Order-7-3 quasiregular rhombic tiling
Order-7 truncated triangular tiling	6.7.7 2 7 | 3 [7,3]	Order-3 heptakis heptagonal tiling
Order-7 triangular tiling	37 7 | 3 2 [7,3]	Order-3 heptagonal tiling
Small rhombitriheptagonal tiling	3.4.7.4 3 | 7 2 [7,3]	Deltoidal triheptagonal tiling
Great rhombitriheptagonal tiling	4.7.12 2 7 3 | [7,3]	Order-3 bisected heptagonal tiling
Order-3 snub heptagonal tiling	3.3.3.3.7 | 7 3 2 [7,3]	Order-7-3 floret pentagonal tiling

[edit] The [5,4] group family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual tilings
Order-4 pentagonal tiling	5.5.5.5 4 | 5 2 [5,4]	Order-5 square tiling
Truncated pentagonal tiling	4.10.10 2 4 | 5 [5,4]	Order-5 tetrakis square tiling
tetrapentagonal tiling	4.5.4.5 2 | 5 4 [5,4]	Order-5-4 quasiregular rhombic tiling
Order-5 truncated square tiling	8.8.5 2 5 | 4 [5,4]	Order-4 pentakis pentagonal tiling
Order-5 square tiling	45 5 | 4 2 [5,4]	Order-4 pentagonal tiling
Small rhombitetrapentagonal tiling	4.4.5.4 4 | 5 2 [5,4]	Deltoidal tetrapentagonal tiling
Great rhombitetrapentagonal tiling	4.8.10 2 5 4 | [5,4]	Order-4 bisected pentagonal tiling
Order-4 snub pentagonal tiling	3.3.4.3.5 | 5 4 2 [5,4]	Order-5-4 floret pentagonal tiling

[edit] (4 3 3) family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual tilings
Order-4-3-3_t0 tiling	(3.4)^3 3 | 3 4 (4 3 3)	Order-4-3-3_t0 dual tiling
Order-4-3-3_t01 tiling	3.8.3.8 3 3 | 4 (4 3 3)	Order-4-3-3_t01 dual tiling
Order-4-3-3_t12 tiling	3.6.4.6 4 3 | 3 (4 3 3)	Order-4-3-3_t12 dual tiling
Order-4-3-3_t2 tiling	(3.3)4 4 | 3 3 (4 3 3)	Order-4-3-3_t2 dual tiling
Order-4-3-3_t012 tiling	6.6.8 4 3 3 | (4 3 3)	Order-4-3-3_t012 dual tiling
Order-4-3-3_snub tiling	3.3.3.3.3.4 | 4 3 3 (4 3 3)	Order-4-3-3_snub dual tiling

[edit] See also

• Convex uniform honeycomb - The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.

[edit] External links

• Uniform Tessellations on the Euclid plane
• Tessellations of the Plane
• David Bailey's World of Tessellations
• k-uniform tilings
• n-uniform tilings

[edit] References

• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. 
• H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401-50.