A ''k''-uniform tiling is a tiling of
tilings of the plane by convex
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 ...
s, connected edge-to-edge, with ''k'' types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tiling can be defined by its
. Higher ''k''-uniform tilings are listed by their vertex figures, but are not generally uniquely identified this way.
The complete lists of ''k''-uniform tilings have been enumerated
up to ''k''=6. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings, and 673 6-uniform tilings. This article lists all solutions up to ''k''=5.
![Halfshift square tiling](https://upload.wikimedia.org/wikipedia/commons/d/d8/Halfshift_square_tiling.svg)
Other
tilings of regular polygons that are not edge-to-edge allow different sized polygons, and continuous shifting positions of contact.
Classification
Such periodic tilings of convex polygons may be classified by the number of
orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
of vertices, edges and tiles. If there are orbits of vertices, a tiling is known as -uniform or -
isogonal; if there are orbits of tiles, as -
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
; if there are orbits of edges, as -
isotoxal
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...
.
''k''-uniform tilings with the same vertex figures can be further identified by their
wallpaper group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...
symmetry.
Enumeration
1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number ''m'' of distinct vertex figures, which are also called ''m''-Archimedean tilings.
Finally, if the number of types of vertices is the same as the uniformity (''m'' = ''k'' below), then the tiling is said to be ''
Krotenheerdt''. In general, the uniformity is greater than or equal to the number of types of vertices (''m'' ≥ ''k''), as different types of vertices necessarily have different orbits, but not vice versa. Setting ''m'' = ''n'' = ''k'', there are 11 such tilings for ''n'' = 1; 20 such tilings for ''n'' = 2; 39 such tilings for ''n'' = 3; 33 such tilings for ''n'' = 4; 15 such tilings for ''n'' = 5; 10 such tilings for ''n'' = 6; and 7 such tilings for ''n'' = 7.
1-uniform tilings (regular)
A tiling is said to be ''regular'' if the
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 ...
of the tiling
acts transitively
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
on the ''flags'' of the tiling, where a flag is a triple consisting of a mutually incident
vertex
Vertex, vertices or vertexes may refer to:
Science and technology Mathematics and computer science
*Vertex (geometry), a point where two or more curves, lines, or edges meet
*Vertex (computer graphics), a data structure that describes the position ...
, edge and tile of the tiling. This means that, for every pair of flags, there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an
edge-to-edge tiling
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
by
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
regular polygons. There must be six
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
s, four
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
s or three regular
hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
s at a vertex, yielding the three regular tessellations.
m-Archimedean and k-uniform tilings
Vertex-transitivity means that for every pair of vertices there is a
symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...
mapping the first vertex to the second.
If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as ''Archimedean'', ''
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
'' or ''demiregular'' tilings. Note that there are two
mirror image
A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
(enantiomorphic or
chiral
Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is distinguishable from ...
) forms of 3
4.6 (snub hexagonal) tiling, only one of which is shown in the following table. All other regular and semiregular tilings are achiral.
Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.
1-uniform tilings (semiregular)
2-uniform tilings
There are twenty (20) 2-uniform tilings of the Euclidean plane. (also called 2-
isogonal tilings or
demiregular tiling In geometry, the ''demiregular tilings'' are a set of Euclidean tessellations made from 2 or more regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral ...
s)
Vertex types are listed for each. If two tilings share the same two vertex types, they are given subscripts 1,2.
3-uniform tilings
There are 61 3-uniform tilings of the Euclidean plane. 39 are 3-Archimedean with 3 distinct vertex types, while 22 have 2 identical vertex types in different symmetry orbits.
3-uniform tilings, 3 vertex types
3-uniform tilings, 2 vertex types (2:1)
4-uniform tilings
There are 151 4-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 33 4-uniform tilings with 4 distinct vertex types, as well as finding 85 of them with 3 vertex types, and 33 with 2 vertex types.
4-uniform tilings, 4 vertex types
There are 33 with 4 types of vertices.
4-uniform tilings, 3 vertex types (2:1:1)
There are 85 with 3 types of vertices.
4-uniform tilings, 2 vertex types (2:2) and (3:1)
There are 33 with 2 types of vertices, 12 with two pairs of types, and 21 with 3:1 ratio of types.
5-uniform tilings
There are 332 5-uniform tilings of the Euclidean plane. Brian Galebach's search identified 332 5-uniform tilings, with 2 to 5 types of vertices. There are 74 with 2 vertex types, 149 with 3 vertex types, 94 with 4 vertex types, and 15 with 5 vertex types.
5-uniform tilings, 5 vertex types
There are 15 5-uniform tilings with 5 unique vertex figure types.
5-uniform tilings, 4 vertex types (2:1:1:1)
There are 94 5-uniform tilings with 4 vertex types.
5-uniform tilings, 3 vertex types (3:1:1) and (2:2:1)
There are 149 5-uniform tilings, with 60 having 3:1:1 copies, and 89 having 2:2:1 copies.
5-uniform tilings, 2 vertex types (4:1) and (3:2)
There are 74 5-uniform tilings with 2 types of vertices, 27 with 4:1 and 47 with 3:2 copies of each.
There are 29 5-uniform tilings with 3 and 2 unique vertex figure types.
Higher k-uniform tilings
''k''-uniform tilings have been enumerated up to 6. There are 673 6-uniform tilings of the Euclidean plane. Brian Galebach's search reproduced Krotenheerdt's list of 10 6-uniform tilings with 6 distinct vertex types, as well as finding 92 of them with 5 vertex types, 187 of them with 4 vertex types, 284 of them with 3 vertex types, and 100 with 2 vertex types.
References
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*
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*
*
*
* Order in Space: A design source book, Keith Critchlow, 1970
* Chapter X: The Regular Polytopes
*
*
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* Dale Seymour and
Jill Britton, ''Introduction to Tessellations'', 1989, , pp. 50–57
External links
Euclidean and general tiling links:
n-uniform tilings Brian Galebach
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{{Tessellation
Euclidean plane geometry
Regular tilings
Tessellation