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Truncated hexagonal tiling
 
 
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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 truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagon
Dodecagon

In geometry, a dodecagon is any polygon with 12 sides and twelve angles....
s (12-sides) and one triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 on each vertex
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....
.

As the name implies this tiling is constructed by a truncation
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....
 operation applies to a 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....
, leaving dodecagons in place of the original hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s, and new triangles at the original vertex locations. It is given an extended 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 t0,1.

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 truncated hextille, constructed as a truncation
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....
 operation applied to a 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....
 (hextille).

Related polyhedra and tilings
This tiling is 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 (3.2n.2n), and continues 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...
.






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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 truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagon
Dodecagon

In geometry, a dodecagon is any polygon with 12 sides and twelve angles....
s (12-sides) and one triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 on each vertex
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....
.

As the name implies this tiling is constructed by a truncation
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....
 operation applies to a 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....
, leaving dodecagons in place of the original hexagon
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
s, and new triangles at the original vertex locations. It is given an extended 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 t0,1.

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 truncated hextille, constructed as a truncation
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....
 operation applied to a 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....
 (hextille).

Related polyhedra and tilings


This tiling is 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 (3.2n.2n), and continues 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...
. These vertex-transitive
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....
 figures have (*n32) reflectional symmetry.
Triangular Prism

(3.4.4)
Triangular prism

In geometry, a triangular prism or three-sided prism is a type of Prism ; it is a polyhedron made of a triangle base, a Translation copy, and 3 faces joining corresponding sides....

(*232)
Uniform Polyhedron 33 T01

(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....

(*332)
Uniform Polyhedron 43 T01

(3.8.8)
Truncated cube

The truncated cube, or truncated hexahedron, is an Archimedean solid. It has 6 regular octagonal faces, 8 regular triangle faces, 24 vertices and 36 edges....

(*432)
Uniform Polyhedron 53 T01

(3.10.10)
Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....

(*532)
Uniform Polyhedron 63 T01

(3.12.12)
(*632)
(3.14.14)
(*732)


There are 3 regular and 8 semiregular tilings in the plane.

There is only one 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....
 of a truncated hexagonal tiling. (Naming the colors by indices around a vertex: 122.) The tiling colors shown in the table is a mixture of 3 types of colored-vertices (a 3-uniform coloring).

See also

  • Tilings of regular polygons
  • List of uniform tilings


External links