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; these 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 its group notation: the sequence of the reflection orders of the fundamental domain vertices.
A fundamental domain triangle is denoted (p q r), where p, q, r are whole numbers > 1, i.e. ≥ 2; a fundamental domain right triangle is denoted (p q 2). 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 several symbolic schemes for denoting these figures: The modified Schläfli symbol for a right triangle domain: (p q 2) → {p, q}. The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, then the graph is linear, since diagram nodes with connectivity 2 are not connected to each other by a diagram branch (since domain mirrors meeting at 90 degrees generate no new mirrors). The Wythoff symbol takes the three integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite to a domain vertex, then the reflection order of this domain vertex is given before the bar. Finally, a uniform tiling can be described by its vertex configuration: the (identical) sequence of polygons around each (equivalent) 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 and vertices). 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 integer reflection orders, including:
+ Euclidean planeOrbifoldsymmetryCoxeter groupCoxeterdiagramNotes Compact *333 (3 3 3) [3] 3 reflective forms, 1 snub *442 (4 4 2) [4,4] 5 reflective forms, 1 snub *632 (6 3 2) [6,3] 7 reflective forms, 1 snub *2222 (∞ 2 ∞ 2) × [∞,2,∞] 3 reflective forms, 1 snub Noncompact (Frieze) *∞∞ (∞) [∞] *22∞ (2 2 ∞) × [∞,2] 2 reflective forms, 1 snub
+ Hyperbolic planeOrbifoldsymmetryCoxeter groupCoxeterdiagramNotes Compact *pq2 (p q 2) [p,q] 2(p+q) < pq *pqr (p q r) [(p,q,r)] pq+pr+qr < pqr, i.e. + + < 1 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, (2 2 2 2), is represented by two sets of parallel mirrors, which in general can make a rectangular fundamental domain. It generates no new tilings.
A further prismatic symmetry group, (∞ 2 2), 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.trianglesParentTruncatedRectifiedBitruncatedBirectified(dual)CantellatedOmnitruncated(Cantitruncated)SnubWythoff symbol q p 2 2 q p 2 p q 2 p q p q 2 p q 2 p q 2 p q 2Schläfli 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 diagramVertex config.pqq.2p.2p(p.q)2p.2q.2qqpp.4.q.44.2p.2q3.3.p.3.qSquare tiling(4 4 2){4,4}4.8.84.4.4.44.8.8{4,4}4.4.4.44.8.83.3.4.3.4Hexagonal tiling(6 3 2){6,3}3.12.123.6.3.66.6.6{3,6}3.4.6.44.6.123.3.3.3.6
General fundamental triangles: (p q r) Wythoff symbol(p q r)Fund.triangles q p r r q p r p q r p q p q r p q r p q r p q rCoxeter diagramVertex 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.pTriangular(3 3 3)(3.3)33.6.3.6(3.3)33.6.3.6(3.3)33.6.3.66.6.63.3.3.3.3.3
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 by 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 are fundamental domains that place vertices at infinity, such as (∞ 2 3), etc.
Right angle fundamental triangles: (p q 2) (p q 2)Fund.trianglesParentTruncatedRectifiedBitruncatedBirectified(dual)CantellatedOmnitruncated(Cantitruncated)SnubWythoff symbol q p 2 2 q p 2 p q 2 p q p q 2 p q 2 p q 2 p q 2Schläfli symbolt{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 diagramVertex config.pqq.2p.2pp.q.p.qp.2q.2qqpp.4.q.44.2p.2q3.3.p.3.q(5 4 2)V4.8.10{5,4}4.10.104.5.4.55.8.8{4,5}4.4.5.44.8.103.3.4.3.5(5 5 2)V4.10.10{5,5}5.10.105.5.5.55.10.10{5,5}5.4.5.44.10.103.3.5.3.5(7 3 2)V4.6.14{7,3}3.14.143.7.3.77.6.6{3,7}3.4.7.44.6.143.3.3.3.7(8 3 2)V4.6.16{8,3}3.16.163.8.3.88.6.6{3,8}3.4.8.44.6.163.3.3.3.8
General fundamental triangles: (p q r) Wythoff symbol(p q r)Fund.triangles q p r r q p r p q r p q p q r p q r p q r p q rCoxeter diagramVertex config.(p.r)qr.2p.q.2p(p.q)rq.2r.p.2r(q.r)pr.2q.p.2q2p.2q.2r3.r.3.q.3.p(4 3 3)V6.6.8(3.4)33.8.3.8(3.4)33.6.4.6(3.3)43.6.4.66.6.83.3.3.3.3.4(4 4 3)V6.8.8(3.4)43.8.4.8(4.4)33.6.4.6(3.4)44.6.4.66.8.83.3.3.4.3.4(4 4 4)V8.8.8(4.4)44.8.4.8(4.4)44.8.4.8(4.4)44.8.4.88.8.83.4.3.4.3.4
Expanded lists of uniform tilings
There are several 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. Zigzags (apeirogons alternating between two angles) can also be used. 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 ∞).
Branko Grünbaum and G. C. Shephard, in the 1987 book Tilings and patterns, section 12.3, enumerate a list of 25 uniform tilings, including the 11 convex forms, and add 14 more they call hollow tilings, using the first two expansions above: star polygon faces and generalized vertex figures.Tiles and Patterns, Table 12.3.1, p. 640
H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, in the 1954 paper 'Uniform polyhedra', Table 8: Uniform Tessellations, use the first three expansions and enumerate a total of 38 uniform tilings. If a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings.
In 1981, Grünbaum, Miller, and Shephard, in their paper Uniform Tilings with Hollow Tiles, list 25 tilings using the first two expansions and 28 more when the third is added (making 53 using Coxeter et al.'s definition). When the fourth is added, they list an additional 23 uniform tilings and 10 families (8 depending on continuous parameters and 2 on discrete parameters).
Besides the 11 convex solutions, the 28 uniform star tilings listed by Coxeter et al., grouped by shared edge graphs, are shown below, followed by 15 more listed by Grünbaum et al. that meet Coxeter et al.'s definition but were missed by them.
This set is not proved complete. By "2.25" is meant tiling 25 in Grünbaum et al.'s table 2 from 1981.
The following three tilings are exceptional in that there is only finitely many of one face type: two apeirogons in each. Sometimes the order-2 apeirogonal tiling is not included, as its two faces meet at more than one edge. + Frieze group symmetryMcNeillDiagramVertexConfig.WythoffSymmetryNotesI1 ∞.∞ p1m1 (Two half-plane tiles, order-2 apeirogonal tiling)I2 4.4.∞ ∞ 2 2 p1m1 Apeirogonal prismI33.3.3.∞ 2 2 ∞ p11g Apeirogonal antiprism
For clarity, the tilings are not colored from here onward (due to the overlaps). A set of polygons around one vertex is highlighted. McNeill only lists tilings given by Coxeter et al. (1954). The eleven convex uniform tilings have been repeated for reference.
Wallpaper group symmetryMcNeillJim McNeillGrünbaum et al., 1981EdgediagramHighlightedVertexConfig.WythoffSymmetryConvex1.94.4.4.4 4 2 4 p4mI42.144.∞.4/3.∞4.∞.−4.∞ 4/3 4 ∞ p4mConvex1.24 6.6.63 2 6 p6mConvex1.253.3.3.3.3.36 2 3 p6mI52.26(3.∞.3.∞.3.∞)/2 3/2 3 ∞p3m1Convex1.233.6.3.62 3 6 p6mI62.25 6.∞.6/5.∞6.∞.−6.∞ 6/5 6 ∞p6mI72.24 ∞.3.∞.3/23.∞.−3.∞ 3/2 3 ∞p6mConvex1.14 3.4.6.4 3 6 2 p6m11.153/2.12.6.12−3.12.6.12 3/2 6 6p6m1.164.12.4/3.12/114.12.−4.−12 2 6 (3/2 6/2) p6mConvex1.5 4.8.8 2 4 4 p4m22.7 4.8/3.∞.8/3 4 ∞ 4/3p4m1.7 8/3.8.8/5.8/78.8/3.−8.−8/3 4/3 4 (4/2 ∞/2) p4m2.6 8.4/3.8.∞−4.8.∞.8 4/3 ∞ 4p4mConvex1.20 3.12.12 2 3 6 p6m32.17 6.12/5.∞.12/5 6 ∞ 6/5p6m1.21 12/5.12.12/7.12/1112.12/5.−12.−12/5 6/5 6 (6/2 ∞/2) p6m2.16 12.6/5.12.∞−6.12.∞.12 6/5 ∞ 6p6m41.18 12/5.3.12/5.6/53.12/5.−6.12/5 3 6 6/5p6m1.19 12/5.4.12/7.4/34.12/5.−4.−12/5 2 6/5 (3/2 6/2) p6m1.17 4.3/2.4.6/53.−4.6.−4 3/2 6 2p6m52.5 8.8/3.∞ 4/3 4 ∞ p4m62.15 12.12/5.∞ 6/5 6 ∞ p6m71.6 8.4/3.8/54.−8.8/3 2 4/3 4 p4mConvex1.11 4.6.12 2 3 6 p6m81.13 6.4/3.12/74.−6.12/5 2 3 6/5 p6m91.12 12.6/5.12/76.−12.12/5 3 6/5 6 p6m101.8 4.8/5.8/5−4.8/3.8/3 2 4 4/3p4m111.22 12/5.12/5.3/2−3.12/5.12/5 2 3 6/5p6mConvex1.1 3.3.3.4.4 non-Wythoffian cmm121.2 4.4.3/2.3/2.3/23.3.3.−4.−4 non-WythoffiancmmConvex1.3 3.3.4.3.4 2 4 4 p4g131.4 4.3/2.4.3/2.3/23.3.−4.3.−4 2 4/3 4/3 p4g142.4 3.4.3.4/3.3.∞3.4.3.−4.3.∞ 4/3 4 ∞ p4Convex1.10 3.3.3.3.6 2 3 6 p6 2.1 3/2.∞.3/2.∞.3/2.4/3.4/33.4.4.3.∞.3.∞ non-Wythoffian cmm2.2 3/2.∞.3/2.∞.3/2.4.43.−4.−4.3.∞.3.∞ non-Wythoffian cmm 2.3 3/2.∞.3/2.4.4.3/2.4/3.4/33.4.4.3.−4.−4.3.∞ non-Wythoffian p3 2.8 4.∞.4/3.8/3.84.8.8/3.−4.∞ non-Wythoffian p4m2.9 4.∞.4/3.8.8/3−4.8.8/3.4.∞ non-Wythoffian p4m 2.10 4.∞.4/3.8.4/3.84.8.−4.8.−4.∞ non-Wythoffian p4m2.11 4.∞.4/3.8.4/3.84.8.−4.8.−4.∞ non-Wythoffian p4g2.12 4.∞.4/3.8/3.4.8/34.8/3.4.8/3.−4.∞ non-Wythoffian p4m2.13 4.∞.4/3.8/3.4.8/34.8/3.4.8/3.−4.∞ non-Wythoffian p4g 2.18 3/2.∞.3/2.4/3.4/3.3/2.4/3.4/33.4.4.3.4.4.3.∞ non-Wythoffian p6m2.19 3/2.∞.3/2.4.4.3/2.4.43.−4.−4.3.-4.−4.3.∞ non-Wythoffian p6m2.20 3/2.∞.3/2.∞.3/2.12/11.6.12/113.12.−6.12.3.∞.3.∞ non-Wythoffian p6m2.21 3/2.∞.3/2.∞.3/2.12.6/5.123.−12.6.−12.3.∞.3.∞ non-Wythoffian p6m2.22 3/2.∞.3/2.∞.3/2.12/7.6/5.12/73.12/5.6.12/5.3.∞.3.∞ non-Wythoffian p6m2.23 3/2.∞.3/2.∞.3/2.12/5.6.12/53.−12/5.−6.−12/5.3.∞.3.∞ non-Wythoffian p6m