# Uniform tiling

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In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

## Contents

Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere.

Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices.

A fundamental domain triangle is (pqr), and a right triangle (pq 2), where p, q, r are whole numbers greater than 1. The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q and r.

There are a number of symbolic schemes for naming these figures, from a modified Schläfli symbol for right triangle domains: (pq 2) → {p, q}. The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, the graph is linear since order-2 domain nodes generate no reflections. The Wythoff symbol takes the 3 integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite a domain node, it is given before the bar.

Finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex.

All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by Norman Johnson are called truncation (cutting vertices), rectification (cutting vertices until edges disappear), and cantellation (cutting edges). Omnitruncation is an operation that combines truncation and cantellation. Snubbing is an operation of alternate truncation of the omnitruncated form. (See Uniform polyhedron#Wythoff construction operators for more details.)

## Coxeter groups

Coxeter groups for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams:

For groups with whole number orders, including:

Euclidean plane
Orbifold
symmetry
Coxeter group Coxeter
diagram
notes
Compact
*333(3 3 3)${\displaystyle {\tilde {A}}_{2}}$[3[3]]3 reflective forms, 1 snub
*442(4 4 2)${\displaystyle {\tilde {B}}_{2}}$[4,4]5 reflective forms, 1 snub
*632(6 3 2)${\displaystyle {\tilde {G}}_{2}}$[6,3]7 reflective forms, 1 snub
*2222( 2 2)${\displaystyle {\tilde {I}}_{1}}$ × ${\displaystyle {\tilde {I}}_{1}}$[,2,]3 reflective forms, 1 snub
Noncompact (frieze)
*()${\displaystyle {\tilde {I}}_{1}}$[]
*22(2 2 )${\displaystyle {\tilde {I}}_{1}}$ × ${\displaystyle {\tilde {A}}_{2}}$[,2]2 reflective forms, 1 snub
Hyperbolic plane
Orbifold
symmetry
Coxeter group Coxeter
diagram
notes
Compact
*pq2(p q 2)[p,q]2(p+q) < pq
*pqr(p q r)[(p,q,r)]pq+pr+qr < pqr
Paracompact
*p2(p 2)[p,]p>=3
*pq(p q )[(p,q,)]p,q>=3, p+q>6
*p(p )[(p,,)]p>=3
*()[(,,)]

## Uniform tilings of the Euclidean plane

There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles.

These symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the semiregular tilings are repeated from different symmetry constructors.

A prismatic symmetry group represented by (2 2 2 2) represents by two sets of parallel mirrors, which in general can have a rectangular fundamental domain. It generates no new tilings.

A further prismatic symmetry group represented by ( 2 2) which has an infinite fundamental domain. It constructs two uniform tilings, the apeirogonal prism and apeirogonal antiprism.

The stacking of the finite faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane. It is called the elongated triangular tiling, composed of alternating layers of squares and triangles.

Right angle fundamental triangles: (pq 2)

(pq 2)Fund.
triangles
ParentTruncatedRectifiedBitruncatedBirectified
(dual)
CantellatedOmnitruncated
(Cantitruncated)
Snub
Wythoff symbol q|p 22 q|p2 |pq2 p|qp|q 2pq| 2pq 2 ||pq 2
Schläfli symbol {p,q}t{p,q}r{p,q}2t{p,q}=t{q,p}2r{p,q}={q,p}rr{p,q}tr{p,q}sr{p,q}
Coxeter diagram
Vertex config. pqq.2p.2p(p.q)2p. 2q.2qqpp. 4.q.44.2p.2q3.3.p. 3.q
Square tiling
(4 4 2)

{4,4}

4.8.8

4.4.4.4

4.8.8

{4,4}

4.4.4.4

4.8.8

3.3.4.3.4
Hexagonal tiling
(6 3 2)

{6,3}

3.12.12

3.6.3.6

6.6.6

{3,6}

3.4.6.4

4.6.12

3.3.3.3.6

General fundamental triangles: (p q r)

Wythoff symbol
(p q r)
Fund.
triangles
q | p rr q | pr | p qr p | qp | q rp q | rp q r || p q r
Coxeter diagram
Vertex config. (p.q)rr.2p.q.2p(p.r)qq.2r.p. 2r(q.r)pq.2r.p. 2rr.2q.p. 2q3.r.3.q.3.p
Triangular
(3 3 3)

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

(3.3)3

3.6.3.6

6.6.6

3.3.3.3.3.3

Non-simplical fundamental domains

The only possible fundamental domain in Euclidean 2-space that is not a simplex is the rectangle (∞ 2 ∞ 2), with Coxeter diagram: . All forms generated from it become a square tiling.

## Uniform tilings of the hyperbolic plane

There are infinitely many uniform tilings of convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r).

A sampling is shown here with a Poincaré disk projection.

The Coxeter-Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.

Further symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains starting with (2 2 2 3), etc., that can generate new forms. As well there's fundamental domains that place vertices at infinity, such as ( 2 3), etc.

Right angle fundamental triangles: (pq 2)

(p q 2)Fund.
triangles
ParentTruncatedRectifiedBitruncatedBirectified
(dual)
CantellatedOmnitruncated
(Cantitruncated)
Snub
Wythoff symbol q | p 22 q | p2 | p q2 p | qp | q 2p q | 2p q 2 || p q 2
Schläfli symbol t{p,q}t{p,q}r{p,q}2t{p,q}=t{q,p}2r{p,q}={q,p}rr{p,q}tr{p,q}sr{p,q}
Coxeter diagram
Vertex figure pq(q.2p.2p)(p.q.p.q)(p. 2q.2q)qp(p. 4.q.4)(4.2p.2q)(3.3.p. 3.q)
(5 4 2)
V4.8.10

{5,4}

4.10.10

4.5.4.5

5.8.8

{4,5}

4.4.5.4

4.8.10

3.3.4.3.5
(5 5 2)
V4.10.10

{5,5}

5.10.10

5.5.5.5

5.10.10

{5,5}

5.4.5.4

4.10.10

3.3.5.3.5
(7 3 2)
V4.6.14

{7,3}

3.14.14

3.7.3.7

7.6.6

{3,7}

3.4.7.4

4.6.14

3.3.3.3.7
(8 3 2)
V4.6.16

{8,3}

3.16.16

3.8.3.8

8.6.6

{3,8}

3.4.8.4

4.6.16

3.3.3.3.8

General fundamental triangles (p q r)

Wythoff symbol
(p q r)
Fund.
triangles
q | p rr q | pr | p qr p | qp | q rp q | rp q r || p q r
Coxeter diagram
Vertex figure (p.r)q(r.2p.q.2p)(p.q)r(q.2r.p. 2r)(q.r)p(r.2q.p. 2q)(2p.2q.2r)(3.r.3.q.3.p)
(4 3 3)
V6.6.8

(3.4)3

3.8.3.8

(3.4)3

3.6.4.6

(3.3)4

3.6.4.6

6.6.8

3.3.3.3.3.4
(4 4 3)
V6.8.8

(3.4)4

3.8.4.8

(4.4)3

3.6.4.6

(3.4)4

4.6.4.6

6.8.8

3.3.3.4.3.4
(4 4 4)
V8.8.8

(4.4)4

4.8.4.8

(4.4)4

4.8.4.8

(4.4)4

4.8.4.8

8.8.8

3.4.3.4.3.4

## 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 polygon 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. If a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.

Besides the 11 convex solutions, the 28 uniform star tilings listed by Coxeter et al., grouped by shared edge graphs, are shown below. For clarity, apeirogons are not coloured in the first seven tilings, and thereafter only the polygons around one vertex are coloured.

This set is not proved complete.

Frieze group symmetry
# [1] DiagramVertex
Config
WythoffSymmetryNotes
I1 .p1m1(Two half-plane tiles, order-2 apeirogonal tiling)
I2 4.4. 2 | 2p1m1 Apeirogonal prism
I3 3.3.3.| 2 2 p11g Apeirogonal antiprism
Wallpaper group symmetry
McNeill [1] Grünbaum [2] Edge
diagram
SolidVertex
Config
WythoffSymmetry
I4 4..4/3.
4..-4.
4/3 4 |p4m
I5 (3..3..3.)/23/2 | 3 p6m
I6 6..6/5.
6..-6.
6/5 6 |
I7 .3..3/2
.3..-3
3/2 3 |
115 3/2.12.6.12
-3.12.6.12
3/2 6 | 6p6m
16 4.12.4/3.12/11
4.12.4/3.-12
2 6 (3/2 6/2) |
2 8/3.4.8/3.4 | 4/3p4m
7 8/3.8.8/5.8/7
8/3.8.-8/3.-8
4/3 4 (4/2 /2) |
8.4/3.8.
8.-4.8.
4/3 | 4
3 12/5.6.12/5.6 | 6/5p6m
21 12/5.12.12/7.12/11
12/5.12.-12/5.-12
6/5 6 (6/2 /2) |
12.6/5.12.
12.-6.12.
6/5 | 6
418 12/5.3.12/5.6/53 6 | 6/5p6m
19 12/5.4.12/7.4/3
12/5.4.-12/5.-4
2 6/5 (3/2 6/2) |
17 4.3/2.4.6/5
4.-3.4.-6
3/2 6 | 2
5 8.8/3.4/3 4 |p4m
6 12.12/5.6/5 6 |p6m
76 8.4/3.8/5
4.8.-8/3
2 4/3 4 |p4m
813 6.4/3.12/7
-6.4.12/5
2 3 6/5 |p6m
912 12.6/5.12/7
-12.6.12/5
3 6/5 6 |p6m
108 4.8/5.8/5
-4.8/3.8/3
2 4 | 4/3p4m
1122 12/5.12/5.3/2
12/5.12/5.-3
2 3 | 6/5p6m
122 4.4.3/2.3/2.3/2
4.4.-3.-3.-3
non-Wythoffian cmm
134 4.3/2.4.3/2.3/2
4.-3.4.-3.-3
| 2 4/3 4/3p4g
14 3.4.3.4/3.3.
3.4.3.-4.3.
| 4/3 4 p4g

## Self-dual tilings

Tilings can also be self-dual. The square tiling, with Schläfli symbol {4,4}, is self-dual; shown here are two square tilings (red and black), dual to each other.

## Uniform tilings using star polygons

Seeing a star polygon as a nonconvex polygon with twice as many sides allows star polygons, and counting these as regular polygons allows them to be used in a uniform tiling. These polygons are labeled as {Nα} for a isotoxal nonconvex 2N-gon with external dihedral angle α. Its external vertices are labeled as N*
α
, and internal N**
α
. This expansion to the definition requires corners with only 2 polygons to not be considered vertices. The tiling is defined by its vertex configuration as a cyclic sequence of convex and nonconvex polygons around every vertex. There are 4 such uniform tilings with adjustable angles α, and 18 uniform tilings that only work with specific angles; yielding a total of 22 uniform tilings that use star polygons. [3]

All of these tilings are topologically related to the ordinary uniform tilings with convex regular polygons, with 2-valence vertices ignored, and square faces as digons, reduced to a single edge.

 3.6*α.6**αTopological 3.12.12 4.4*α.4**αTopological 4.8.8 6.3*α.3**αTopological 6.6.6 3.3*α.3.3**αTopological 3.6.3.6
 4.6.4*π/6.6Topological 4.4.4.4 (8.4*π/4)2Topological 4.4.4.4 12.12.4*π/3Topological 4.8.8 3.3.8*π/12.4**π/3.8*π/12Topological 4.8.8 3.3.8*π/12.3.4.3.8*π/12Topological 4.8.8 3.4.8.3.8*π/12Topological 4.8.8 5.5.4*4π/10.5.4*π/10Topological 3.3.4.3.4 4.6*π/6.6**π/2.6*π/6Topological 6.6.6 (4.6*π/6)3Topological 6.6.6 9.9.6*4π/9Topological 6.6.6 (6.6*π/3)2Topological 3.6.3.6 (12.3*π/6)2Topological 3.6.3.6 3.4.6.3.12*π/6Topological 4.6.12 3.3.3.12*π/6.3.3.12*π/6Topological 3.12.12 18.18.3*2π/9Topological 3.12.12 3.6.6*π/3.6Topological 3.4.6.4 8.3*π/12.8.6*5π/12Topological 3.4.6.4 9.3.9.3*π/9Topological 3.6.3.6

### Uniform tilings using alternating polygons

Star polygons of the form {pα} can also represent convex 2p-gons alternating two angles, the simplest being a rhombus {2α}. Allowing these as regular polygons, creates more uniform tilings, with some example below.

 3.2*.6.2**Topological 3.4.6.4 4.4.4.4Topological 4.4.4.4 (2*π/6.2**π/3)2Topological 4.4.4.4 2*π/6.2*π/6.2**π/3.2**π/3Topological 4.4.4.4 4.2*π/6.4.2**π/3Topological 4.4.4.4

## Related Research Articles

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in Schwarz (1873).

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6}.

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex.

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular hexagonal tiling and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.

In geometry, the Wythoff symbol is a notation representing a Wythoff construction of a uniform polyhedron or plane tiling within a Schwarz triangle. It was first used by Coxeter, Longuet-Higgins and Miller in their enumeration of the uniform polyhedra. Later the Coxeter diagram was developed to mark uniform polytopes and honeycombs in n-dimensional space within a fundamental simplex.

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.

The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied by to get 180 degrees. An unlabeled branch implicitly represents order-3, and each pair of nodes that is not connected by a branch at all represents a pair of mirrors at order-2.

In geometry, a polytope, or a tiling, is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.

In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of tr{7,3}.

In hyperbolic geometry, a uniform hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder.

In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle,, defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles. Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror. Up to 3 face types exist centered on the fundamental triangle corners. Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling.

## References

1. Tiles and Patterns, Table 12.3.1 p.640
2. Tilings and Patterns Branko Gruenbaum, G.C. Shephard, 1987. 2.5 Tilings using star polygons, pp.82-85.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Grünbaum, Branko; Shephard, G. C. (1987). . W. H. Freeman and Company. ISBN   0-7167-1193-1. (Star tilings section 12.3)
• H. S. M. Coxeter, M. S. Longuet-Higgins, J. C. P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 40150 JSTOR   91532 (Table 8)
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$${\displaystyle {\tilde {C}}_{n-1}}$${\displaystyle {\tilde {B}}_{n-1}}$${\displaystyle {\tilde {D}}_{n-1}}$${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
E10Uniform 10-honeycomb{3[11]}δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21