List of convex uniform tilings

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An example uniform tiling as a church floor tiling in Sevilla, Spain

This table shows the 11 convex uniform tilings (regular and semiregular) 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.

John Conway calls the uniform duals Archimedean tilings, in parallel to the Archimedean solid polyhedra.

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 tilings, using star polygons, and reverse orientation vertex configurations.

Laves tilings[edit]

In the 1987 book, Tilings and Patterns, Branko Grünbaum calls the vertex-uniform tilings Archimedean in parallel to the Archimedean solids. Their dual tilings are called Laves tilings in honor of crystallographer Fritz Laves.[1] [2] They're also called Shubnikov–Laves tilings after Shubnikov, Alekseĭ Vasilʹevich.[3] John Conway calls the uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.

The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons. This includes the 3 regular tiles (triangle, square and pentagon) and 8 irregular ones.[4] Each vertex has edges evenly spaced around it. Three dimensional analogues of the planigons are called stereohedrons.

These 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 four triangles, and two corners containing eight triangles.

Eleven planigons
Triangles Quadrilaterals Pentagons Hexagon
Alchemy fire symbol.svg
V6.6.6
CDel node.pngCDel split1.pngCDel branch.png
Tiling face 4-8-8.svg
V4.8.8
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Tiling face 4-6-12.svg
V4.6.12
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Tiling face 3-12-12.svg
V3.12.12
CDel 2.png
Regular quadrilateral.svg
V4.4.4.4
CDel labelinfin.pngCDel branch.pngCDel 2.pngCDel branch.pngCDel labelinfin.png
Tiling face 3-6-3-6.svg
V3.6.3.6
CDel 2.png
Tiling face 3-4-6-4.svg
V3.4.6.4
CDel 2.png
Tiling face 3-3-4-3-4.svg
V3.3.4.3.4
CDel 2.png
Tiling face 3-3-3-3-6.svg
V3.3.3.3.6
CDel 2.png
Tiling face 3-3-3-4-4.svg
V3.3.3.4.4
CDel 2.png
Hexagon.svg
V3.3.3.3.3.3
CDel 2.png

Convex uniform tilings of the Euclidean plane[edit]

All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon one of three Schwarz triangle (4,4,2), (6,3,2), or (3,3,3), with symmetry represented by Coxeter groups: [4,4], [6,3], or [3[3]]. Alternated forms such as the snub can also be represented by special markups within each system. Only one uniform tiling can't be constructed by a Wythoff process, but can be made by an elongation of the triangular tiling. An orthogonal mirror construction [∞,2,∞] also exists, seen as two sets of parallel mirrors making a rectangular fundamental domain. If the domain is square, this symmetry can be doubled by a diagonal mirror into the [4,4] family.

Families:

  • (4,4,2), , [4,4] - Symmetry of the regular square tiling
    • , [∞,2,∞]
  • (6,3,2), , [6,3] - Symmetry of the regular hexagonal tiling and triangular tiling.
    • (3,3,3), , [3[3]]

The [4,4] group family[edit]

Uniform tilings
(Platonic and Archimedean)
Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram(s)
Dual-uniform tilings
(called Laves or Catalan tilings)
1-uniform n5.svg
Square tiling (quadrille)
Square tiling vertfig.pngRegular quadrilateral.svg
4.4.4.4 (or 44)
4 | 2 4
p4m, [4,4], (*442)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png
1-uniform 5 dual.svg
self-dual (quadrille)
1-uniform n2.svg
Truncated square tiling (truncated quadrille)
Truncated square tiling vertfig.pngTiling face 4-8-8.svg
4.8.8
2 | 4 4
4 4 2 |
p4m, [4,4], (*442)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png or CDel node 1.pngCDel split1-44.pngCDel nodes 11.png
1-uniform 2 dual.svg
Tetrakis square tiling (kisquadrille)
1-uniform n9.svg
Snub square tiling (snub quadrille)
Snub square tiling vertfig.pngTiling face 3-3-4-3-4.svg
3.3.4.3.4
| 4 4 2
p4g, [4+,4], (4*2)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png or CDel node h.pngCDel split1-44.pngCDel nodes hh.png
1-uniform 9 dual.svg
Cairo pentagonal tiling (4-fold pentille)

The [6,3] group family[edit]

Platonic and Archimedean tilings Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram(s)
Dual Laves tilings
1-uniform n1.svg
Hexagonal tiling (hextille)
Hexagonal tiling vertfig.pngAlchemy fire symbol.svg
6.6.6 (or 63)
3 | 6 2
2 6 | 3
3 3 3 |
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel split1.pngCDel branch 11.png
1-uniform 1 dual.svg
Triangular tiling (deltille)
1-uniform n7.svg
Trihexagonal tiling (hexadeltille)
Trihexagonal tiling vertfig.pngTiling face 3-6-3-6.svg
(3.6)2
2 | 6 3
3 3 | 3
p6m, [6,3], (*632)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel branch 10ru.pngCDel split2.pngCDel node 1.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
1-uniform 7 dual.svg
Rhombille tiling (rhombille)
1-uniform n4.svg
Truncated hexagonal tiling (truncated hextille)
Truncated hexagonal tiling vertfig.pngTiling face 3-12-12.svg
3.12.12
2 3 | 6
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
1-uniform 4 dual.svg
Triakis triangular tiling (kisdeltille)
1-uniform n11.svg
Triangular tiling (deltille)
Triangular tiling vertfig.pngHexagon.svg
3.3.3.3.3.3 (or 36)
6 | 3 2
3 | 3 3
| 3 3 3
p6m, [6,3], (*632)
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node 1.pngCDel split1.pngCDel branch.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel split1.pngCDel branch hh.png
1-uniform 11 dual.svg
Hexagonal tiling (hextille)
1-uniform n6.svg
Rhombitrihexagonal tiling (rhombihexadeltille)
Small rhombitrihexagonal tiling vertfig.pngTiling face 3-4-6-4.svg
3.4.6.4
3 | 6 2
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
1-uniform 6 dual.svg
Deltoidal trihexagonal tiling (tetrille)
1-uniform n3.svg
Truncated trihexagonal tiling (truncated hexadeltille)
Great rhombitrihexagonal tiling vertfig.pngTiling face 4-6-12.svg
4.6.12
2 6 3 |
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
1-uniform 3 dual.svg
Kisrhombille tiling (kisrhombille)
1-uniform 10.png
Snub trihexagonal tiling (snub hextille)
Snub hexagonal tiling vertfig.pngTiling face 3-3-3-3-6.svg
3.3.3.3.6
| 6 3 2
p6, [6,3]+, (632)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
1-uniform 10 dual.svg
Floret pentagonal tiling (6-fold pentille)

Non-Wythoffian uniform tiling[edit]

Platonic and Archimedean tilings Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram
Dual Laves tilings
1-uniform n8.svg
Elongated triangular tiling (isosnub quadrille)
Tiling 33344-vertfig.pngTiling face 3-3-3-4-4.svg
3.3.3.4.4
2 | 2 (2 2)
cmm, [∞,2+,∞], (2*22)
CDel node.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
CDel node h.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
1-uniform 8 dual.svg
Prismatic pentagonal tiling (iso(4-)pentille)

Uniform colorings[edit]

There are a total of 32 uniform colorings of the 11 uniform tilings:

  1. Triangular tiling - 9 uniform colorings, 4 wythoffian, 5 nonwythoffian
    • Uniform tiling 63-t2.pngUniform tiling 333-t1.pngUniform tiling 333-snub.pngUniform tiling 63-h12.pngUniform triangular tiling 111222.pngUniform triangular tiling 112122.pngUniform triangular tiling 111112.pngUniform triangular tiling 111212.pngUniform triangular tiling 111213.png
  2. Square tiling - 9 colorings: 7 wythoffian, 2 nonwythoffian
    • Square tiling uniform coloring 1.pngSquare tiling uniform coloring 2.pngSquare tiling uniform coloring 7.pngSquare tiling uniform coloring 8.pngSquare tiling uniform coloring 3.pngSquare tiling uniform coloring 6.pngSquare tiling uniform coloring 4.pngSquare tiling uniform coloring 5.pngSquare tiling uniform coloring 9.png
  3. Hexagonal tiling - 3 colorings, all wythoffian
    • Uniform tiling 63-t0.pngUniform tiling 63-t12.pngUniform tiling 333-t012.png
  4. Trihexagonal tiling - 2 colorings, both wythoffian
    • Uniform polyhedron-63-t1.pngUniform tiling 333-t01.png
  5. Snub square tiling - 2 colorings, both alternated wythoffian
    • Uniform tiling 44-h01.pngUniform tiling 44-snub.png
  6. Truncated square tiling - 2 colorings, both wythoffian
    • Uniform tiling 44-t12.pngUniform tiling 44-t012.png
  7. Truncated hexagonal tiling - 1 coloring, wythoffian
    • Uniform tiling 63-t01.png
  8. Rhombitrihexagonal tiling - 1 coloring, wythoffian
    • Uniform tiling 63-t02.png
  9. Truncated trihexagonal tiling - 1 coloring, wythoffian
    • Uniform tiling 63-t012.png
  10. Snub hexagonal tiling - 1 coloring, alternated wythoffian
    • Uniform tiling 63-snub.png
  11. Elongated triangular tiling - 3 coloring, nonwythoffian
    • Elongated triangular tiling 1.png

See also[edit]

References[edit]

  1. ^ Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. pp. 59, 96. ISBN 0-7167-1193-1. 
  2. ^ The Symmetries of things, Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations, p. 288
  3. ^ Encyclopaedia of Mathematics: Orbit - Rayleigh Equation edited by Michiel Hazewinkel, 1991
  4. ^ Ivanov, A. B. (2001), "Planigon", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 

Further reading[edit]

External links[edit]