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List of uniform planar tilings

 

 
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List of uniform planar tilings
 
 
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This table shows the 11 convex uniform tiling
Uniform tessellation

In mathematics, a uniform tessellation is a tessellation of a d-dimensional space, or a surface, such that all its Vertex-transitive, i.e., there is the same combination and arrangement of faces at each vertex....
s of the Euclidean plane
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, and their dual tilings.

There are three regular, and eight semiregular, tilings
Tiling by regular polygons

Plane Tessellation by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler in Harmonices Mundi....
 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 polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
, 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. Different Symmetry can be expressed on the same geometric figure with the Face following different uniform color patterns....
s.






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This table shows the 11 convex uniform tiling
Uniform tessellation

In mathematics, a uniform tessellation is a tessellation of a d-dimensional space, or a surface, such that all its Vertex-transitive, i.e., there is the same combination and arrangement of faces at each vertex....
s of the Euclidean plane
Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Greek mathematics Euclid of Alexandria. Euclid's Elements is the earliest known systematic discussion of geometry....
, and their dual tilings.

There are three regular, and eight semiregular, tilings
Tiling by regular polygons

Plane Tessellation by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler in Harmonices Mundi....
 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 polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
, 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. Different Symmetry can be expressed on the same geometric figure with the Face following different uniform color patterns....
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 forms, using star polygon
Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
s, and reverse orientation vertex configuration
Vertex configuration

In polyhedral geometry a vertex configuration is a short-hand notation for representing a polyhedron vertex figure as the sequence of faces around a vertex....
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 Gr?nbaum 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 set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s, and the dual tilings Laves tilings in honor of crystalographer Fritz Laves.

Convex uniform tilings of the Euclidean plane


The R3 [4,4] group family


Platonic and Archimedean tilingsVertex 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....

Wythoff symbol(s)
Wythoff symbol

In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....

Symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 Laves tilings

The V3 [6,3] group family


Platonic and Archimedean tilingsVertex 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....

Wythoff symbol(s)
Wythoff symbol

In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....

Symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 Laves tilings


Non-Wythoffian uniform tiling


Platonic and Archimedean tilingsVertex 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....

Wythoff symbol(s)
Wythoff symbol

In geometry, a Wythoff symbol is a short-hand notation, created by mathematician Willem Abraham Wythoff, for naming the regular and semiregular polyhedra using a Wythoff construction, by representing them as tilings on the surface of a sphere, Euclidean plane, or hyperbolic plane....

Symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 Laves tilings


See also

  • Convex uniform honeycomb
    Convex uniform honeycomb

    In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
     - The 28 uniform 3-dimensional tessellations, a parallel construction to the convex uniform Euclidean plane tilings.
  • Uniform tilings in hyperbolic plane
    Uniform tilings in hyperbolic plane

    There are an infinite number of uniform tilings on the hyperbolic plane based on the where p + q + r > 9 List_of_regular_polytopes#Hyperbolic_tilings....


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