Hexagonal tiling

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Hexagonal tiling

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol

Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
of or t (as a truncated triangular tiling).

Conway

John Horton Conway

John Horton Conway is a prolific mathematician active in the theory of finite group , knot theory, number theory, combinatorial game theory and coding theory....
calls it a hextille.

The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the 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....
and the triangular tiling

Triangular tiling

Honeycomb

A honeycomb is a mass of hexagonal waxcells built by honey bees in their beehive to contain their larva and stores of honey and pollen.beekeeping may remove the entire honeycomb to harvest honey....
, and various crystal lattices.

e are 3 distinct uniform coloring

Uniform coloring

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Encyclopedia

In geometry

Geometry

Schläfli symbol

John Horton Conway

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....
and the triangular tiling

Triangular tiling

Honeycomb

Uniform colorings

There are 3 distinct 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 of a hexagonal tiling. (Naming the colors by indices on the 3 hexagons around a vertex: 111, 112, 123.)

The 3 colorings, named by their generating Wythoff symbols and symmetry

List of planar symmetry groups

This article summarizes the classes of Discrete space planar symmetry groups:#2 infinite set of point groups#7 Frieze groups#17 wallpaper groups...
are:

3 | 6 2
*p632 (p6m)

2 6 | 3
*p632 (p6m)

3 3 3 |
*333 (p3)

Related polyhedra and tilings

The hexagonal tiling can be stretched and adjusted to other geometric proportions and different symmetries. For example, the standard brick pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted on the two long edges, dividing them into two colinear edges.

This tiling is topologically related as a part of sequence of regular polyhedra with 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....
(n³), and continue into the hyperbolic plane

Hyperbolic plane

In mathematics, the term hyperbolic plane may refer to:* A two-dimensional quadratic space with a non-singular isotropic quadratic form* A plane in hyperbolic geometry...
.

(3³

Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
)

(4³

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
)

(5³

Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
)

(6³) tiling

(7³) tiling

Order-3 heptagonal tiling

In geometry, the order-3 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schl?fli symbol of .The image shows a Poincar? disk model projection of the hyperbolic plane....

It is also topologically related as a part of sequence of uniform truncated

Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
polyhedra with vertex figure (n.6.6).

(3.6.6)

Truncated tetrahedron

The truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangle faces, 12 vertices and 18 edges....

(4.6.6)

Truncated octahedron

The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces, 6 Square faces, 24 vertices and 36 edges. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

(5.6.6)

Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....

(6.6.6) tiling

(7.6.6) tiling

Order-7 truncated triangular tiling

In geometry, the Order 7 truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex ....

Wythoff constructions from hexagonal and triangular tilings

Like the uniform polyhedra

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....
there are eight uniform tilings

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....
that can be based from the regular hexagonal tiling (or the dual 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....
).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Tiling	Schläfli symbol Schläfli symbol In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....	Wythoff symbol Wythoff construction File:Wythoffian_construction_diagram.pngIn geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling....	Vertex figure 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....
Hexagonal tiling	t₀	3 \| 6 2	6³
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 ....	t_0,1	2 3 \| 6	3.12.12
Rectified hexagonal tiling (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 .... )	t₁	2 \| 6 3	(3.6)²
Bitruncated hexagonal tiling (Truncated triangular tiling)	t_1,2	2 6 \| 3	6.6.6
Dual hexagonal tiling (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.... )	t₂	6 \| 3 2	3⁶
Cantellated hexagonal tiling (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 .... )	t_0,2	6 3 \| 2	3.4.6.4
Omnitruncated hexagonal tiling (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 .... )	t_0,1,2	6 3 2 \|	4.6.12
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 ....	s	\| 6 3 2	3.3.3.3.6

Hexagonal tiling

Uniform colorings

Related polyhedra and tilings

Wythoff constructions from hexagonal and triangular tilings

See also

External links