Tiling by regular polygons

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Tiling by regular polygons

Plane

Plane (mathematics)

In mathematics, a plane is a curvature surface. Planes can arise as subspaces of some higher dimensional space, as with the walls of a room, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry....
tilings
Tessellation

A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of the parts of the plane or of other surfaces....
by regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler

Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
in Harmonices Mundi.

Regular tilings
Following 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....
and Shephard (section 1.3), a tiling is said to be regular if the 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....
of the tiling acts transitively

Group action

Vertex (geometry)

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Encyclopedia

Plane

Plane (mathematics)

Johannes Kepler

Regular tilings

Following Grünbaum

Branko Grünbaum

Symmetry group

Group action

Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
, 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

Congruence (geometry)

In geometry, two sets of point are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translation s, rotations and reflection s....
regular polygons. There must be six equilateral triangle

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also Equiangular polygon; that is, all three internal angles are also congruent to each other and are each 60?....
s, four square

Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
s or three regular hexagon

Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s at a vertex, yielding the three regular tessellations.

3⁶ Triangular tiling Triangular tiling In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....	4⁴ Square tiling Square tiling In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....	6³ Hexagonal tiling Hexagonal tiling In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....

Archimedean, uniform or semiregular tilings

Vertex-transitivity

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
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
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....
or semiregular tilings. Note that there are two mirror image

Mirror Image

"Mirror Image" is an episode of the television series The Twilight Zone ....
(enantiomorphic or chiral

Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
) forms of 3⁴.6 (snub hexagonal) tiling, both of which are shown in the following table. All other regular and semiregular tilings are achiral.

3⁴.6 Snub hexagonal tiling Snub hexagonal tiling In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex ....	3⁴.6 Snub hexagonal tiling Snub hexagonal tiling In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex .... reflection Reflection Reflection or reflexion may refer to:...	3.6.3.6 Trihexagonal tiling Trihexagonal tiling In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....
3³.4² Elongated triangular tiling Elongated triangular tiling In geometry, the elongated triangular tiling is a Tiling by regular polygons of the Euclidean plane. There are three triangles and two squares on each vertex ....	3².4.3.4 Snub square tiling Snub square tiling In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....	3.4.6.4 Small rhombitrihexagonal tiling Small rhombitrihexagonal tiling In geometry, the small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two Square s, and one hexagon on each vertex ....
4.8² Truncated square tiling Truncated square tiling In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....	3.12² Truncated hexagonal tiling Truncated hexagonal tiling In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex ....	4.6.12 Great rhombitrihexagonal tiling Great rhombitrihexagonal tiling In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex ....

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

In geometry, an interior angle is an angle formed by two sides of a simple polygon that share an endpoint, namely, the angle on the inner side of the polygon....
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 eleven of these can occur in a uniform 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 (cannot appear in any tiling of regular polygons)
3.8.24 (cannot appear in any tiling of regular polygons)
3.9.18 (cannot appear in any tiling of regular polygons)
3.10.15 (cannot appear in any tiling of regular polygons)
3.12² - semi-regular, truncated hexagonal tiling
Truncated hexagonal tiling

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex ....
4.5.20 (cannot appear in any tiling of regular polygons)
4.6.12 - semi-regular, great rhombitrihexagonal tiling
Great rhombitrihexagonal tiling

In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex ....
4.8² - semi-regular, truncated square tiling
Truncated square tiling

In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex . This is the only edge-to-edge tiling by regular convex polygons which contains an octagon....
5².10 (cannot appear in any tiling of regular polygons)
6³ - regular, hexagonal tiling
Hexagonal tiling

In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of or t .John Horton Conway calls it a hextille....

With 4 polygons at a vertex:

3².4.12 - not uniform
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
, has two different types of vertices 3².4.12 and 3⁶
3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
3².6² - not uniform, occurs in two patterns with vertices 3².6²/3⁶ and 3².6²/3.6.3.6.
3.6.3.6 - semi-regular, trihexagonal tiling
Trihexagonal tiling

In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex ....
4⁴ - regular, square tiling
Square tiling

In geometry, the Square tiling is a regular tiling of the Euclidean plane. It has Schl?fli symbol of .John Horton Conway calls it a quadrille....
3.4².6 - not uniform, has vertices 3.4².6 and 3.6.3.6.
3.4.6.4 - semi-regular, small rhombitrihexagonal tiling
Small rhombitrihexagonal tiling

In geometry, the small rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two Square s, and one hexagon on each vertex ....

With 5 polygons at a vertex:

3⁴.6 - snub hexagonal tiling
Snub hexagonal tiling

In geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex ....
3³.4² - semi-regular, Elongated triangular tiling
Elongated triangular tiling

In geometry, the elongated triangular tiling is a Tiling by regular polygons of the Euclidean plane. There are three triangles and two squares on each vertex ....
3².4.3.4 - semi-regular, Snub square tiling
Snub square tiling

In geometry, the snub square tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex . It has Schl?fli symbol of s....

With 6 polygons at a vertex:

3⁶ - regular, Triangular tiling
Triangular tiling

In geometry, the triangular tiling is one of the three regular tessellations of the Euclidean plane. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees....

Other edge-to-edge tilings

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

3².6² and 3⁶	3².6² and 3.6.3.6
3².4.12 and 3⁶	3.4².6 and 3.6.3.6

Such periodic tilings may be classified by the number of orbits

Group action

In algebra and geometry, a group action is a way of describing symmetry of objects using group . The essential elements of the object are described by a Set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformation of the set....
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.

The hyperbolic plane

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

Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two dimensional space, for any given line l and point P not on l, there is exactly one line through P th...
. 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 (Using Poincaré disc model projection)