Tiling by regular polygons

Plane tilings Tessellation

A tessellation or tiling of the plane [i] is a collection of plane figure [i]s that fills th ...

by regular polygon Regular polygon

A regular polygon is a simple polygon [i] which is [i] and equilateral [i] ...

s have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler Johannes Kepler

Johannes Kepler , a key figure in the scientific revolution [i], was a German [i] mathematician [i] ...

in Harmonices Mundi.

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Encyclopedia

Plane tilings Tessellation

A tessellation or tiling of the plane [i] is a collection of plane figure [i]s that fills th ...

by regular polygon Regular polygon

A regular polygon is a simple polygon [i] which is [i] and equilateral [i] ...

s have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler Johannes Kepler

Johannes Kepler , a key figure in the scientific revolution [i], was a German [i] mathematician [i] ...

in Harmonices Mundi.

Regular tilings

Following Grünbaum and Shephard , a tiling is said to be regular if the symmetry group Symmetry group

The symmetry [i] group of an object is the group [i] of all isometries [i] under which it is invariant [i] ...

of the tiling acts transitively on the flags of the tiling, where a flag is a triple consisting of a mutually incident vertex, 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 by congruent regular polygons. There must be six equilateral triangle Triangle

A triangle is one of the basic shape [i]s of geometry [i]: a polygon [i] with three vertices [i] ...

s, four squares or three regular hexagon Hexagon

In geometry [i], a hexagon is a polygon [i] with six edge [i]s and six vertices [i]. ...

s at a vertex, yielding the three regular tessellations.

3.3.3.3.3.3
Triangular tiling Triangular tiling

In geometry [i], the triangular tiling is a regular tiling [i] of the Euclidean plane....

4.4.4.4
Square tiling

6.6.6
Hexagonal tiling Hexagonal tiling

In geometry [i], the hexagonal tiling is a regular tiling of the Euclidean plane. ...

Archimedean, uniform or semiregular tilings

Vertex-transitivity means that for every pair of vertices there is a symmetry operation 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 or semiregular tilings.

3.3.3.3.6 Snub hexagonal tiling Snub hexagonal tiling In geometry [i], the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. ...	3.6.3.6 Trihexagonal tiling Trihexagonal tiling In geometry [i], the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
3.3.3.4.4 Elongated triangular tiling	3.3.4.3.4 Snub square tiling
3.4.6.4 Small rhombitrihexagonal tiling Small rhombitrihexagonal tiling In geometry [i], the Small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane [i]. ...	4.8.8 Truncated square tiling Truncated square tiling In geometry [i], the truncated square tiling is a semiregular tiling of the Euclidean plane. ...
3.12.12 Truncated hexagonal tiling Truncated hexagonal tiling In geometry [i], the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ...	4.6.12 Great rhombitrihexagonal tiling Great rhombitrihexagonal tiling In geometry [i], the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...

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.

Combinations of regular polygons that can meet at a vertex

The internal angle Internal angle

[i]...

s of the polygons meeting at a vertex must add to 360 degrees. A regular -gon has internal angle degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex. Only fifteen of these can occur in any tiling of regular polygons. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.

With 3 polygons at a vertex:

3.7.42
3.8.24
3.9.18
3.10.15
3.12.12
4.5.20
4.6.12
4.8.8
5.5.10
6.6.6

With 4 polygons at a vertex:

3.3.4.12 or 3.4.3.12
3.3.6.6 or 3.6.3.6
4.4.4.4
3.4.4.6 or 3.4.6.4

With 5 polygons at a vertex:

3.3.3.3.6
3.3.3.4.4 or 3.3.4.3.4

With 6 polygons at a vertex:

3.3.3.3.3.3

Other edge-to-edge tilings

Any number of non-uniform edge-to-edge tilings by regular polygons may be drawn. Here are four examples:

3.3.6.6 & 3.3.3.3.3.3	3.3.6.6 & 3.6.3.6
3.3.4.12 & 3.3.3.3.3.3	3.4.4.6 & 3.6.3.6

Such periodic tilings may be classified by the number of orbits 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; if there are orbits of edges, as -isotoxal. The examples above are four of the twenty 2-uniform tilings. Chavey lists all those edge-to-edge tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal.

Tilings that are not edge-to-edge

Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may also be known as uniform if they are vertex-transitive; there are eight families of such uniform tilings, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles.

Beyond the plane

These tessellations are also related to regular and semiregular polyhedra and tessellations of the hyperbolic plane Hyperbolic geometry

Hyperbolic geometry is a non-Euclidean geometry [i], meaning that the parallel postulate [i] of Euclidean geometry [i] ...

. Semiregular polyhedra are made from regular polygon faces, but their angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry have angles smaller than they do in the plane. In both these cases, that the arrangement of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive.

Some regular tilings of the hyperbolic plane