List of uniform planar tilings
Encyclopedia
This table shows the 11 convex uniform tiling
s (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.
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 coloring
s. 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 nonconvex tilings, using star polygons, and reverse orientation vertex configuration
s.
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 4 triangles, and two corners containing 8 triangles.
In the 1987 book, Tilings and Patterns, Branko Grünbaum
calls the vertex uniform tilings Archimedean in parallel to the Archimedean solid
s, and the dual tilings Laves tilings in honor of crystallographer
Fritz Laves
. John Conway
calls the duals Catalan tilings, in parallel to the Catalan solid
polyhedra.
Uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane...
s (regular and semiregular) of the Euclidean plane
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, and their dual tilings.
There are three regular, and eight semiregular, tilings
Tiling by regular polygons
Plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in Harmonices Mundi.- Regular tilings :...
in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face.
Uniform tilings are listed by their vertex configuration
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...
, 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 coloring
Uniform coloring
In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive...
s. 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 nonconvex tilings, using star polygons, and reverse orientation vertex configuration
Vertex configuration
In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...
s.
Dual tilings are listed by their face configuration
Face configuration
In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....
, the number of faces at each vertex of a face. For example V4.8.8 means isosceles triangle tiles with one corner with 4 triangles, and two corners containing 8 triangles.
In the 1987 book, Tilings and Patterns, Branko Grünbaum
Branko Grünbaum
Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
calls the vertex uniform tilings Archimedean in parallel to the Archimedean solid
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
s, and the dual tilings Laves tilings in honor of crystallographer
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
Fritz Laves
Fritz Laves
-References:...
. John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
calls the duals Catalan tilings, in parallel to the Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
polyhedra.
The [4,4] group family
| Uniform tilings (Platonic and Archimedean) |
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... Wythoff symbol(s) Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation... |
Dual Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another... uniform tilings (called Laves or Catalan tilings) |
|---|
The [6,3] group family
| Platonic and Archimedean tilings | Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... Wythoff symbol(s) Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation... |
Dual Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another... Laves tilings |
|---|
Non-Wythoffian uniform tiling
| Platonic and Archimedean tilings | Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:... Wythoff symbol(s) Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation... |
Dual Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another... Laves tilings |
|---|
See also
- Uniform tilingUniform tilingIn geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane...
- Convex uniform honeycombConvex uniform honeycombIn geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
- The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings. - Uniform tilings in hyperbolic planeUniform tilings in hyperbolic planeThere are an infinite number of uniform tilings on the hyperbolic plane based on the where 1/p + 1/q + 1/r ...