List of Euclidean uniform tilings

File:Semi-regular-floor-3464.JPG

An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling

File:Brückner Vielflache Fig. 96b - 98.jpg

Uniform tilings and their duals drawn by Max Brückner in Vielecke und Vielflache (1900)

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 called these uniform duals Catalan tilings, in parallel to the Catalan 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 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases the parameter is continuous, and in the other 2 it is discrete. The set is not known to be complete.

Laves tilings

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. They're also called Shubnikov–Laves tilings after Aleksei Shubnikov. John Conway called 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 hexagon) and 8 irregular ones. 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. The orientations of the vertex planigons (up to D₁₂) are consistent with the vertex diagrams in the below sections.

Eleven planigons

Triangles				Quadrilaterals			Pentagons			Hexagon
File:Tiling 6 dual face.svg V63 Template:CDD	File:Tiling truncated 4a dual face.svg V4.82 Template:CDD	File:Tiling great rhombi 3-6 dual face.svg V4.6.12 Template:CDD	File:Tiling truncated 6 dual face.svg V3.122 Template:CDD	File:Tiling 4a dual face.svg V44 Template:CDD	File:Tiling 3-6 dual face.svg V(3.6)2 Template:CDD	File:Tiling small rhombi 3-6 dual face.svg V3.4.6.4 Template:CDD	File:Tiling snub 4-4 left dual face.svg V32.4.3.4 Template:CDD	File:Tiling snub 3-6 left dual face.svg V34.6 Template:CDD	File:Tiling elongated 3 dual face.svg V33.42 Template:CDD	File:Tiling 3 dual face.svg V36 Template:CDD

Convex uniform tilings of the Euclidean plane

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), ${\widetilde{BC}}_2$, [4,4] – Symmetry of the regular square tiling
  • ${\tilde{I}}_1^2$, [∞,2,∞]
• (6,3,2), ${\tilde{G}}_2$, [6,3] – Symmetry of the regular hexagonal tiling and triangular tiling.
  • (3,3,3), ${\tilde{A}}_2$, [3^[3]]

The [4,4] group family

Template:Uniform tiling list tableTemplate:Uniform tiling list tableTemplate:Uniform tiling list table

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)

The [6,3] group family

Template:Uniform tiling list tableTemplate:Uniform tiling list tableTemplate:Uniform tiling list tableTemplate:Uniform tiling list tableTemplate:Uniform tiling list tableTemplate:Uniform tiling list tableTemplate:Uniform tiling list table

Platonic and Archimedean tilings	Vertex figure and dual face Wythoff symbol(s) Symmetry group Coxeter diagram(s)	Dual Laves tilings

Non-Wythoffian uniform tiling

Template:Uniform tiling list table

Platonic and Archimedean tilings	Vertex figure and dual face Wythoff symbol(s) Symmetry group Coxeter diagram	Dual Laves tilings

Uniform colorings

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

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

See also

• Convex uniform honeycomb – The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.
• Euclidean tilings by convex regular polygons
• List of tessellations
• Percolation threshold
• Uniform tilings in hyperbolic plane

Further reading

• Template:The Geometrical Foundation of Natural Structure (book) (Section 2–3 Circle packings, plane tessellations, and networks, pp. 34–40).
• (Casey Mann at the University of Washington)

External links

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