geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a uniform tiling is a

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

of the plane by

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

faces with the restriction of being

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

. Uniform tilings can exist in both the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

and

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

. Uniform tilings are related to the finite

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

which can be considered uniform tilings of the

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

. Most uniform tilings can be made from a

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

starting with a

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...

and a singular generator point inside of the

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

. A planar symmetry group has a polygonal

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 Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

, 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 In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

for right triangle domains: (''p'' ''q'' 2) → . 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 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 pol ...

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 In geometry, a vertex configurationCrystallography ...

, the sequence of polygons around each vertex. All uniform tilings can be constructed from various operations applied to

regular tiling Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Eucl ...

s. These operations as named by Norman Johnson are called

truncation In mathematics and computer science, truncation is limiting the number of digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...

(cutting vertices),

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

(cutting vertices until edges disappear), and

cantellation In geometry, a cantellation is a 2nd-order truncation in any dimension that bevels a regular polytope at its edges and at its vertices, creating a new facet in place of each edge and of each vertex. Cantellation also applies to regular tiling ...

(cutting edges).

Omnitruncation In geometry, an omnitruncation is an operation applied to a regular polytope (or honeycomb) in a Wythoff construction that creates a maximum number of facets. It is represented in a Coxeter–Dynkin diagram with all nodes ringed. It is a ''shortc ...

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 group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

s for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams: For groups with whole number orders, including:

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

s, 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 In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane.Conway (2008), p.263 Thorold Gosset called it a ''2-dimensional semi-check ...

and

apeirogonal antiprism In geometry, an apeirogonal antiprism or infinite antiprismConway (2008), p. 263 is the arithmetic limit of the family of antiprisms; it can be considered an infinite polyhedron or a tiling of the plane. If the sides are equilateral triangles, i ...

. The stacking of the finite faces of these two prismatic tilings constructs one

non-Wythoffian In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

uniform tiling of the plane. It is called the

elongated triangular tiling In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol :e. ...

, composed of alternating layers of squares and triangles. Right angle fundamental triangles: (''p'' ''q'' 2) General fundamental triangles: (p q r) Non-simplical fundamental domains The only possible fundamental domain in Euclidean 2-space that is not a

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

is the rectangle (∞ 2 ∞ 2), with

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

: . All forms generated from it become a

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the s ...

Uniform tilings of the hyperbolic plane

There are infinitely many uniform tilings of convex regular polygons on the

, each based on a different reflective symmetry group (p q r). A sampling is shown here with a

Poincaré disk Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * L ...

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) General fundamental triangles (p q r)

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 In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

tiles can be included. #

Apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the ...

s, , 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 A Pythagorean tiling or two squares tessellation is a tiling of a Euclidean plane by squares of two different sizes, in which each square touches four squares of the other size on its four sides. Many proofs of the Pythagorean theorem are ...

. 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 Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentG. C. Shephard, 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 they call ''hollow tilings'' which included the first two expansions above, star polygon faces and vertex figures. H.S.M. Coxeter,

M. S. Longuet-Higgins Michael Selwyn Longuet-Higgins FRS (8 December 1925 – 26 February 2016) was a mathematician and oceanographer at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, England and Institute for Nonlinea ...

, and J. C. P. Miller, in the 1954 paper 'Uniform polyhedra', in ''Table 8: Uniform Tessellations'', use 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. In 1981, Grünbaum, Miller, and Shephard in their paper ''Uniform Tilings with Hollow Tiles'' listed 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. For clarity, apeirogons are not coloured from here onward. A set of polygons round one vertex is highlighted. McNeill only lists tilings given by Coxeter et al. (1954). The eleven convex uniform tilings have been repeated for reference. There are two uniform tilings for the vertex figure 4.8.-4.8.-4.∞ (Grünbaum et al. 2.10 and 2.11) and also two uniform tilings for the vertex figure 4.8/3.4.8/3.-4.∞ (Grünbaum et al. 2.12 and 2.13), with different symmetries. There is also a third tiling for each vertex figure that is only pseudo-uniform (vertices come in two symmetry orbits). They use different sets of square faces. Hence, for star Euclidean tilings, the vertex figure does not necessarily determine the tiling. In the pictures below, the included squares with horizontal and vertical edges are marked with a central dot. A single square has edges highlighted. Grünbaum ambiguous tilings 1.png, 2.10 and 2.12 (p4m) Grünbaum ambiguous tilings 2.png, 2.11 and 2.13 (p4g) Grünbaum ambiguous tilings 3.png, Pseudo-uniform The tilings with zigzags are listed below. The notation denotes a zigzag with angle 0 < α < π. The apeirogon can be considered as the special case α = π. The symmetries are given for the generic case: there are sometimes special values of α that increase the symmetry. Tilings 3.1 and 3.12 can even become regular; 3.32 already is (it has no free parameters). Sometimes there are special values of α that cause the tiling to degenerate. The tiling pairs 3.17 and 3.18, as well as 3.19 and 3.20, have identical vertex configurations but different symmetries. Tilings 3.7 through 3.10 have the same edge arrangement as 2.1 and 2.2; 3.17 through 3.20 have the same edge arrangement as 2.10 through 2.13; 3.21 through 3.24 have the same edge arrangement as 2.18 through 2.23; and 3.25 through 3.33 have the same edge arrangement as 1.25 (the regular triangular tiling).

Self-dual tilings

Tilings can also be self-dual. The square tiling, with

, 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 In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

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 for a isotoxal nonconvex 2N-gon with external dihedral angle α. Its external vertices are labeled as N, and internal N. This expansion to the definition requires corners with only 2 polygons to not be considered vertices. The tiling is defined by its

vertex configuration In geometry, a vertex configurationCrystallography ...

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

Uniform tilings using alternating polygons

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

References

* Norman Johnson ''Uniform Polytopes'', Manuscript (1991) ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966 * (Star tilings section 12.3) *

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, J. C. P. Miller, ''Uniform polyhedra'', Phil. Trans. 1954, 246 A, 401–50 (Table 8)

External links

*
Uniform Tessellations on the Euclid plane

David Bailey's World of Tessellations

* {{Tessellation