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 ''Archimedean'' in parallel to the Archimedean solids, and the dual tilings ''Laves tilings'' in honor of crystalographer Fritz Laves.

Contents

Convex uniform tilings of the Euclidean plane

The R3 [4,4] group family

The V3 [6,3] group family

Non-Wythoffian uniform tiling

Expanded lists of uniform tilings

Uniform tilings in hyperbolic plane

The [7,3] group family

The [5,4] group family

(4 3 3) family

See also

External links

References

Convex uniform tilings of the Euclidean plane

The R₃ [4,4] group family

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings

The V₃ [6,3] group family

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings

Non-Wythoffian uniform tiling

Platonic and Archimedean tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual Laves tilings

Expanded lists of uniform tilings

There are a number ways the list of uniform tilings can be expanded:
# Vertex figures can have retrograde faces and turn around the vertex more than once.
# Star polygons tiles can be included.
# 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:
# ∞.∞ (Two half-plane tiles, infinite dihedron)
# 4.4.∞ - '∞ 2 | 2' (an infinite prism)
# 3.3.3.∞.∞ '| 2 2 ∞' (an infinite antiprism)
# 4.∞.4/3.∞ '4/3 4 | ∞' (alternate square tiling)
# 3.∞.3.∞.3.∞ - '3/2 | 3 ∞' (alternate triangular tiling)
# 6.∞.6/5.∞ - '6/5 6 | ∞' (alternate trihexagonal tiling with only hexagons)
# ∞.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:
# 'Type 1'
#
★ 3/2.12.6.12 - '3/2 6 | 6'
#
★ 4.12.4/3.12/11 - '2 6 (3/2 3) |'
# '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'
# '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'
# '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'
# 'Type 5'
#
★ 8.8/3.∞ - '4/3 4 ∞ |'
# 'Type 6'
#
★ 12.12/5.∞ - '6/5 6 ∞ |'
# 'Type 7'
#
★ 8.4/3.8/5 - 2 '4/3 4 |'
# 'Type 8'
#
★ 6.4/3.12/7 - '2 3 6/5 |'
# 'Type 9'
#
★ 12.6/5.12/7 - '3 6/5 6 |'
# 'Type 10'
#
★ 4.8/5.8/5 - '2 4 | 4/3'
# 'Type 11'
#
★ 12/5.12/5.3/2 - '2 3 | 6/5'
# 'Type 12'
#
★ 4.4.3/2.3/2.3/2 - non-Wythoffian
# 'Type 13'
#
★ 4.3/2.4.3/2.3/2 - snub
# 'Type 14'
#
★ 3.4.3.4/3.3.∞ - snub

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:

The [7,3] group family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual tilings

The [5,4] group family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual tilings

(4 3 3) family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group	Dual tilings

See also

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

External links

★ Uniform Tessellations on the Euclid plane

★ Tessellations of the Plane

★ David Bailey's World of Tessellations

★ k-uniform tilings

★ n-uniform tilings

References

★ Tilings and Patterns, Grünbaum, Branko; Shephard, G. C., , , W. H. Freeman and Company, 1987, 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.