Uniform tiling

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

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 (p q r), and a right triangle (p q 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: (p q 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			notes
Compact
*333	(3 3 3)	${\tilde {A}}_{2}$	[3^[3]]	3 reflective forms, 1 snub
*442	(4 4 2)	${\tilde {B}}_{2}$	[4,4]	5 reflective forms, 1 snub
*632	(6 3 2)	${\tilde {G}}_{2}$	[6,3]	7 reflective forms, 1 snub
*2222	(∞ 2 ∞ 2)	${\tilde {I}}_{1}$ × ${\tilde {I}}_{1}$	[∞,2,∞]	3 reflective forms, 1 snub
Noncompact (frieze)
*∞∞	(∞)	${\tilde {I}}_{1}$	[∞]
*22∞	(2 2 ∞)	${\tilde {I}}_{1}$ × ${\tilde {A}}_{2}$	[∞,2]	2 reflective forms, 1 snub

Hyperbolic plane
Orbifold symmetry	Coxeter group		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: (p q 2)

(p q 2)	Fund. triangles	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff symbol		p 2	p	p q	q	q 2	2		p q 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.		p^q	q.2p.2p	(p.q)²	p. 2q.2q	q^p	p. 4.q.4	4.2p.2q	3.3.p. 3.q
Square tiling (4 4 2)	0	{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)	0	{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	p r	p	p q	q	q r	r		p q r
Coxeter diagram
Vertex config.		(p.q)^r	r.2p.q.2p	(p.r)^q	q.2r.p. 2r	(q.r)^p	q.2r.p. 2r	r.2q.p. 2q	3.r.3.q.3.p
Triangular (3 3 3)	0	(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: (p q 2)

(p q 2)	Fund. triangles	Parent	Truncated	Rectified	Bitruncated	Birectified (dual)	Cantellated	Omnitruncated (Cantitruncated)	Snub
Wythoff symbol		p 2	p	p q	q	q 2	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		p^q	(q.2p.2p)	(p.q.p.q)	(p. 2q.2q)	q^p	(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	p r	p	p q	q	q r	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:

Vertex figures can have retrograde faces and turn around the vertex more than once.
Star polygon tiles can be included.
Apeirogons, {∞}, can be used as tiling faces.
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.

The vertex figures for the six tilings with convex regular polygons and apeirogon faces.(The Wythoff symbol is given in red.)

Vertex figures for 21 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.

Frieze group symmetry
#[1]	Vertex Config	Wythoff	Symmetry	Notes
I1	∞.∞		p1m1	(Two half-plane tiles, order-2 apeirogonal tiling)
I2	4.4.∞	2	p1m1	Apeirogonal prism
I3	3.3.3.∞	2 2 ∞	p11g	Apeirogonal antiprism

Wallpaper group symmetry
McNeill[1]	Grünbaum[2]	Edge diagram	Solid	Vertex Config	Wythoff	Symmetry
I4				4.∞.4/3.∞ 4.∞.-4.∞	∞	p4m
I5				(3.∞.3.∞.3.∞)/2	3 ∞	p6m
I6				6.∞.6/5.∞ 6.∞.-6.∞	∞
I7				∞.3.∞.3/2 ∞.3.∞.-3	∞
1	15			3/2.12.6.12 -3.12.6.12	6	p6m
1	16			4.12.4/3.12/11 4.12.4/3.-12		p6m
2				8/3.4.8/3.∞	4/3	p4m
	7			8/3.8.8/5.8/7 8/3.8.-8/3.-8
				8.4/3.8.∞ 8.-4.8.∞	4
3				12/5.6.12/5.∞	6/5	p6m
	21			12/5.12.12/7.12/11 12/5.12.-12/5.-12
				12.6/5.12.∞ 12.-6.12.∞	6
4	18			12/5.3.12/5.6/5	6/5	p6m
	19			12/5.4.12/7.4/3 12/5.4.-12/5.-4
	17			4.3/2.4.6/5 4.-3.4.-6	2
5				8.8/3.∞		p4m
6				12.12/5.∞		p6m
7	6			8.4/3.8/5 4.8.-8/3		p4m
8	13			6.4/3.12/7 -6.4.12/5		p6m
9	12			12.6/5.12/7 -12.6.12/5		p6m
10	8			4.8/5.8/5 -4.8/3.8/3	4/3	p4m
11	22			12/5.12/5.3/2 12/5.12/5.-3	6/5	p6m
12	2			4.4.3/2.3/2.3/2 4.4.-3.-3.-3	non-Wythoffian	cmm
13	4			4.3/2.4.3/2.3/2 4.-3.4.-3.-3	2 4/3 4/3	p4g
14				3.4.3.4/3.3.∞ 3.4.3.-4.3.∞	4/3 4 ∞	p4g

Self-dual tilings

The {4,4} square tiling (black) with its dual (red).

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

This example, 4.8^*
_π/8.4^**
_π/4.8^*
_π/4 is considered not edge-to-edge due to the large square, although it can be interpreted as star polygon with pairs of colinear edges.

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 {N_α} 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.[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.

4 uniform tilings with star polygons, angle α
3.6^* _α.6^** _α Topological 3.12.12	4.4^* _α.4^** _α Topological 4.8.8	6.3^* _α.3^** _α Topological 6.6.6	3.3^* _α.3.3^** _α Topological 3.6.3.6

17 uniform tilings with star polygons
4.6.4^* _π/6.6 Topological 4.4.4.4	(8.4^* _π/4)² Topological 4.4.4.4	12.12.4^* _π/3 Topological 4.8.8	3.3.8^* _π/12.4^** _π/3.8^* _π/12 Topological 4.8.8	3.3.8^* _π/12.3.4.3.8^* _π/12 Topological 4.8.8	3.4.8.3.8^* _π/12 Topological 4.8.8
5.5.4^* _4π/10.5.4^* _π/10 Topological 3.3.4.3.4	4.6^* _π/6.6^** _π/2.6^* _π/6 Topological 6.6.6	(4.6^* _π/6)³ Topological 6.6.6	9.9.6^* _4π/9 Topological 6.6.6	(6.6^* _π/3)² Topological 3.6.3.6	(12.3^* _π/6)² Topological 3.6.3.6
3.4.6.3.12^* _π/6 Topological 4.6.12	3.3.3.12^* _π/6.3.3.12^* _π/6 Topological 3.12.12	18.18.3^* _2π/9 Topological 3.12.12	3.6.6^* _π/3.6 Topological 3.4.6.4	8.3^* _π/12.8.6^* _5π/12 Topological 3.4.6.4	9.3.9.3^* _π/9 Topological 3.6.3.6

Uniform tilings using alternating polygons

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

Examples
3.2.6.2* Topological 3.4.6.4	4.4.4.4 Topological 4.4.4.4	(2^* _π/6.2^** _π/3)² Topological 4.4.4.4	2^* _π/6.2^* _π/6.2^ _π/3.2^ _π/3 Topological 4.4.4.4	4.2^* _π/6.4.2^** _π/3 Topological 4.4.4.4

References

Jim McNeill
Tiles and Patterns, Table 12.3.1 p.640
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). Tilings and Patterns. 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, 401–50 JSTOR 91532 (Table 8)

External links

Weisstein, Eric W. "Uniform tessellation". MathWorld.
Uniform Tessellations on the Euclid plane
Tessellations of the Plane
David Bailey's World of Tessellations
k-uniform tilings
n-uniform tilings
Klitzing, Richard. "4D Euclidean tilings".

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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[mcneill-1] Jim McNeill

[2] Tiles and Patterns, Table 12.3.1 p.640

[3] Tilings and Patterns Branko Gruenbaum, G.C. Shephard, 1987. 2.5 Tilings using star polygons, pp.82-85.