Convex uniform honeycomb

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Convex uniform honeycomb

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....
, a convex uniform honeycomb is a uniform space-filling tessellation

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....
in three-dimensional Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
with non-overlapping convex uniform polyhedral

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....
cells.

Twenty-eight such honeycombs exist:

They can be considered the three-dimensional analogue to the uniform tilings of the plane

List of uniform planar tilings

This table shows the 11 convex Uniform tessellations of the Euclidean geometry, and their dual tilings.There are three regular, and eight semiregular, Tiling by regular polygons in the plane....
.

Only 14 of the convex uniform polyhedra appear in these patterns:

set can be called the regular and semiregular honeycombs.

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Encyclopedia

In geometry

Geometry

Tessellation

Euclidean space

Uniform polyhedron

the familiar cubic honeycomb
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
and 7 truncations thereof;
the alternated cubic honeycomb
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
and 4 truncations thereof;
10 prismatic forms based on the uniform plane tilings
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
(11 if including the cubic honeycomb);
5 modifications of some of the above by elongation and/or gyration.

They can be considered the three-dimensional analogue to the uniform tilings of the plane

List of uniform planar tilings

This table shows the 11 convex Uniform tessellations of the Euclidean geometry, and their dual tilings.There are three regular, and eight semiregular, Tiling by regular polygons in the plane....
.

History

1900: Thorold Gosset
Thorold Gosset

Thorold Gosset was an England lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher....
enumerated the list of semiregular convex polytopes with regular cells (Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
1905: Alfredo Andreini
Alfredo Andreini

Alfredo Andreini was an Italian physician and entomologist.He carried out a large collection of insects collected in particular from Cape Verde and in Libya and Eritrea....
enumerated 25 of these tessellations.
1991: Norman Johnson's manuscript Uniform Polytopes identified the complete list of 28.
1994: Branko 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....
, in his paper Uniform tilings of 3-space, also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing
Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedron cells....
. Grünbaum also states that I. Alexeyev of Russia also independently enumerated these forms around the same time.
2006: George Olshevsky
George Olshevsky

George Olshevsky is a freelance editing, writer, publisher, paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List....
, in his manuscript Uniform Panoploid Tetracombs, along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of uniform polychoron
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
s in 4-space).

Only 14 of the convex uniform polyhedra appear in these patterns:

three of the five Platonic solid
Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s,
six of the thirteen Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s, and
five of the infinite family of prism
Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
s.

Names

This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) polyhedra, commonly called Archimedean solid

Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s. Recently 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....
has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations.

The individual honeycombs are listed with names given to them by Norman Johnson

Norman Johnson

Norman W. Johnson is a mathematician, previously at Wheaton College, Massachusetts, Norton, Massachusetts. He earned his Ph.D. from the University of Toronto in 1966 with a dissertation title of The Theory of Uniform Polytopes and Honeycombs under the supervision of H....
. (Some of the terms used below are defined in Uniform polychoron#Geometric derivations

Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
.)

For cross-referencing, they are given with list indices from [A]ndreini (1-22), [W]illiams(1-2,9-19), [J]ohnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and [G]runbaum(1-28).

Compact Euclidean uniform tessellations (by their infinite Coxeter group families)

The fundamental infinite Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
s for 3-space are:

The C^~₃, [4,3,4], cubic, (8 unique forms plus one alternation)
The B^~₃, [4,3^1,1], alternated cubic, (11 forms, 3 new)
The A^~₃ cyclic group, (5 forms, one new)

In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations.

The total unique honeycombs above are 18.

The prismatic stacks from infinite Coxeter groups for 3-space are:

The C^~₂xI^~₁, [4,4]x[8] prismatic group, (2 new forms)
The H^~₂xI^~₁, [6,3]x[8] prismatic group, (7 unique forms)
The A^~₂xI^~₁, [?]x[8] prismatic group, (No new forms)
The I^~₁xI^~₁xI^~₁, [8]x[8]x[8] prismatic group, (These all become a cubic honeycomb)

In addition there is one special elongated form of the triangular prismatic honeycomb.

The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.

Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.

The C^~₃, [4,3,4] group (cubic)

The regular cubic honeycomb, represented by Schläfli symbol , offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.)

Reference Indices	Honeycomb name Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli 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.... symbols	Cell counts/vertex and positions in cubic honeycomb
Reference Indices		(0)	(1)	(2)	(3)	Solids (Partial)	Frames (Perspective)	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....
J_11,15 A₁ W₁ G₂₂	cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... t₀				(8) (4.4.4) 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....			octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
J_12,32 A₁₅ W₁₄ G₇	rectified cubic Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1.... t₁	(2) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....			(4) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....			cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
J₁₃ A₁₄ W₁₅ G₈	truncated cubic Truncated cubic honeycomb The truncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated cubes and octahedron in a ratio of 1:1.... t_0,1	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....			(4) (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....			square pyramid Square pyramid In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
J₁₄ A₁₇ W₁₂ G₉	cantellated cubic Cantellated cubic honeycomb The cantellated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of small rhombicuboctahedron, cuboctahedron, and cubes in a ratio of 1:1:3.... t_0,2	(1) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(2) (4.4.4) 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....		(2) (3.4.4.4)			wedge Wedge (geometry) In geometry, a wedge is a polyhedron solid defined by two triangles and three trapezoid faces. A wedge has five faces, nine edges, and six vertices....
J_11,15	runcinated cubic (same as regular cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... ) t_0,3	(1) (4.4.4) 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....	(3) (4.4.4) 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....	(3) (4.4.4) 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....	(1) (4.4.4) 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....			octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
J₁₆ A₃ W₂ G₂₈	bitruncated cubic Bitruncated cubic honeycomb The Bitruncation cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedron.It is one of 28 Convex uniform honeycomb.... t_1,2	(2) (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....			(2) (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....			(disphenoid tetrahedron)
J₁₇ A₁₈ W₁₃ G₂₅	cantitruncated cubic Cantitruncated cubic honeycomb The cantitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedron, truncated octahedron, and cubes in a ratio of 1:1:3.... t_0,1,2	(1) (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....	(1) (4.4.4) 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....		(2) (4.6.8)			irregular tetrahedron 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....
J₁₈ A₁₉ W₁₉ G₂₀	runcitruncated cubic Runcitruncated cubic honeycomb The runcitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of small rhombicuboctahedron, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3.... t_0,1,3	(1) (3.4.4.4)	(1) (4.4.4) 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....	(2) (4.4.8) Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (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....			oblique trapezoidal pyramid
J₁₉ A₂₂ W₁₈ G₂₇	omnitruncated cubic Omnitruncated cubic honeycomb The omnitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated cuboctahedron and octagonal prism in a ratio of 1:3.... t_0,1,2,3	(1) (4.6.8)	(1) (4.4.8) Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.4.8) Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.6.8)			irregular tetrahedron 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....
J_21,31,51 A₂ W₉ G₁	alternated cubic Tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of alternating octahedron and tetrahedron in a ratio of 1:2.... h₀	(6) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....			(8) (3.3.3) 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....			cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....

B^~₄, h[4,3,4], [4,3^1,1] group

The B^~₄ group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.

The honeycombs from this group are called alternated cubic because the first form can be seen as a cubic honeycomb with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps.

Nodes are indexed left to right as 0,1,0',3 with 0' being below and interchangeable with 0. The alternate cubic names given are based on this ordering.

Referenced indices	Honeycomb name Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	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....
Referenced indices		(0)	(1)	(0')	(3)	Solids (Partial)	Frames (Perspective)
J_21,31,51 A₂ W₉ G₁	alternated cubic Tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of alternating octahedron and tetrahedron in a ratio of 1:2....			(6) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(8) (3.3.3) 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....			cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
J_22,34 A₂₁ W₁₇ G₁₀	truncated alternated cubic Truncated alternated cubic honeycomb The truncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated octahedron, cuboctahedron and truncated tetrahedron in a ratio of 1:1:2....	(1) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....		(2) (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....	(2) (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....
J_12,32 A₁₅ W₁₄ G₇	rectified cubic Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1.... (rectified alternate cubic)	(2) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....		(2) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(2) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....			cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
J_12,32 A₁₅ W₁₄ G₇	rectified cubic Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1.... (cantellated alternate cubic)	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....		(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(4) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....			cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
J₁₆ A₃ W₂ G₂₈	bitruncated cubic Bitruncated cubic honeycomb The Bitruncation cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedron.It is one of 28 Convex uniform honeycomb.... (cantitruncated alternate cubic)	(1) (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....		(1) (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....	(2) (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....			isosceles tetrahedron 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....
J₁₃ A₁₄ W₁₅ G₈	truncated cubic Truncated cubic honeycomb The truncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated cubes and octahedron in a ratio of 1:1.... (bicantellated alternate cubic)	(2) (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....		(2) (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....	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....			square pyramid Square pyramid In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
J_11,15 A₁ W₁ G₂₂	cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... (trirectified alternate cubic)	(4) (4.4.4) 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....		(4) (4.4.4) 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....				octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
J₂₃ A₁₆ W₁₁ G₅	runcinated alternated cubic Runcinated alternated cubic honeycomb The runcinated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of small rhombicuboctahedron, cubes, and tetrahedron in a ratio of 1:1:2....	(1) cube 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....		(3) (3.4.4.4) Rhombicuboctahedron The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....	(1) (3.3.3) 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....
J₁₄ A₁₇ W₁₂ G₉	cantellated cubic Cantellated cubic honeycomb The cantellated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of small rhombicuboctahedron, cuboctahedron, and cubes in a ratio of 1:1:3.... (runcicantellated alternate cubic)	(1) (3.4.4.4)	(2) (4.4.4) 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....	(1) (3.4.4.4)	(1) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....			wedge Wedge (geometry) In geometry, a wedge is a polyhedron solid defined by two triangles and three trapezoid faces. A wedge has five faces, nine edges, and six vertices....
J₂₄ A₂₀ W₁₆ G₂₁	cantitruncated alternated cubic Cantitruncated alternated cubic honeycomb The cantitruncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated cuboctahedron, truncated cube and truncated tetrahedron in a ratio of 1:1:2.... (or runcitruncated alternate cubic)	(1) (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....		(2) (4.6.8) Truncated cuboctahedron The truncated cuboctahedron is an Archimedean solid. It has 12 Square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges....	(1) (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....
J₁₇ A₁₈ W₁₃ G₂₅	cantitruncated cubic Cantitruncated cubic honeycomb The cantitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedron, truncated octahedron, and cubes in a ratio of 1:1:3.... (omnitruncated alternated cubic)	(1) (4.6.8)	(1) (4.4.4) 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....	(1) (4.6.8)	(1) (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....			irregular tetrahedron 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....

A^~₃ group

There are 5 forms constructed from the A^~₃ group, only the quarter cubic honeycomb is unique.

Referenced indices	Honeycomb name Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	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....
Referenced indices		(0)	(1)	(2)	(3)	Solids (Partial)	Frames (Perspective)
J_21,31,51 A₂ W₉ G₁	alternated cubic Tetrahedral-octahedral honeycomb The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of alternating octahedron and tetrahedron in a ratio of 1:2....		(4) (3.3.3) 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....	(6) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(4) (3.3.3) 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....			cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
J_12,32 A₁₅ W₁₄ G₇	rectified cubic Rectified cubic honeycomb The rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of octahedron and cuboctahedron in a ratio of 1:1....	(2) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(2) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....			cuboid Cuboid In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
J_25,33 A₁₃ W₁₀ G₆	quarter cubic Quarter cubic honeycomb The quarter cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedron and truncated tetrahedron in a ratio of 1:1....	(1) (3.3.3) 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....	(1) (3.3.3) 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....	(3) (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....	(3) (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....			Elongated triangular antiprism
J_22,34 A₂₁ W₁₇ G₁₀	truncated alternated cubic Truncated alternated cubic honeycomb The truncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of truncated octahedron, cuboctahedron and truncated tetrahedron in a ratio of 1:1:2....	(1) (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....	(1) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(1) (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....	(2) (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....			Rectangular pyramid
J₁₆ A₃ W₂ G₂₈	bitruncated cubic Bitruncated cubic honeycomb The Bitruncation cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedron.It is one of 28 Convex uniform honeycomb....	(1) (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....	(1) (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....	(1) (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....	(1) (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....			isosceles tetrahedron 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....

Nonwythoffian forms (gyrated and elongated)

Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (gyration) and/or inserting a layer of prisms (elongation).

The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike. In the elongated form, each prism meets a tetrahedron at one triangular end and an octahedron at the other. In the gyroelongated form, prisms that meet tetrahedra at both ends alternate with prisms that meet octahedra at both ends.

The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes.

Referenced indices	symbol	Honeycomb name	cell types (# at each vertex)	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....
J₅₂ A_2' G₂	h:g	gyrated alternated cubic	tetrahedron 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.... (8) octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex.... (6)	triangular orthobicupola Triangular orthobicupola In geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases....
J₆₁ A_? G₃	h:ge	gyroelongated alternated cubic Gyroelongated alternated cubic honeycomb The gyroelongated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of octahedron, triangular prisms, and tetrahedron in a ratio of 1:2:2....	triangular prism 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.... (6) tetrahedron 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.... (4) octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex.... (3)	-
J₆₂ A_? G₄	h:e	elongated alternated cubic Elongated alternated cubic honeycomb The elongated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of octahedron, triangular prisms, and tetrahedron in a ratio of 1:2:2....	triangular prism 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.... (6) tetrahedron 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.... (4) octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex.... (3)
J₆₃ A_? G₁₂	:g x	gyrated triangular prismatic Gyrated triangular prismatic honeycomb The gyrated triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex....	triangular prism 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.... (12)
J₆₄ A_? G₁₅	:ge x	gyroelongated triangular prismatic Gyroelongated triangular prismatic honeycomb The gyroelongated triangular prismatic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is comprised of cubes and triangular prisms in a ratio of 1:2....	triangular prism 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.... (6) cube 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.... (4)

Prismatic stacks

Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings

Tiling by regular polygons

Plane Tessellation by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler in Harmonices Mundi....
, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The 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....
of each is an irregular bipyramid

Bipyramid

An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
whose faces are isosceles triangles.

The C^~₂xI^~₁(∞), [4,4] x [∞], prismatic group

There's only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.

Indices	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli 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.... symbols	Honeycomb name	Plane tiling
J_11,15 A₁ G₂₂	x	Cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... (Square prismatic)	(4.4.4.4) 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....
J₄₅ A₆ G₂₄	t_0,1 x	Truncated/Bitruncated square prismatic Truncated square prismatic honeycomb The truncated square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of octagonal prisms and cubes in a ratio of 1:1....	(4.8.8) 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....
J_11,15 A₁ G₂₂	t₁ x	Cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... (Rectified square prismatic)	(4.4.4.4) 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....
J_11,15 A₁ G₂₂	t_0,2 x	Cubic Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubes. It is an analog of the square tiling of the plane, and part of a dimensional family called hypercube honeycombs.... (Cantellated square prismatic)	(4.4.4.4) 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....
J₄₅ A₆ G₂₄	t_0,1,2 x	Truncated square prismatic Truncated square prismatic honeycomb The truncated square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of octagonal prisms and cubes in a ratio of 1:1.... (Omnitruncated square prismatic)	(4.8.8) 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....
J₄₄ A₁₁ G₁₄	s x	Snub square prismatic Snub square prismatic honeycomb The snub square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of cubes and triangular prisms in a ratio of 1:2....	(3.3.4.3.4) 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....

The H^~₂xI^~₁(∞), [6,3] x [∞] prismatic group

Indices	Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli 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.... symbols	Honeycomb name	Plane tiling
J₄₂ A₅ G₂₆	t₀ x	Hexagonal prismatic Hexagonal prismatic honeycomb The hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space made up of hexagonal prism.It is constructed from a hexagonal tiling extruded into prisms....	(6³) 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....
J₄₆ A₇ G₁₉	t_0,1 x	Truncated hexagonal prismatic Truncated hexagonal prismatic honeycomb The truncated hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of dodecagonal prisms, and triangular prisms in a ratio of 1:2....	(3.12.12) 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 ....
J₄₃ A₈ G₁₈	t₁ x	Trihexagonal prismatic Triangular-hexagonal prismatic honeycomb The triangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of hexagonal prisms and triangular prisms in a ratio of 1:2....	(3.6.3.6) 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 ....
J₄₂ A₅ G₂₆	t_1,2 x	Truncated triangular prismatic Hexagonal prismatic Hexagonal prismatic honeycomb The hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space made up of hexagonal prism.It is constructed from a hexagonal tiling extruded into prisms....	(6.6.6) 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....
J₄₁ A₄ G₁₁	t₂ x	Triangular prismatic Triangular prismatic honeycomb The triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised entirely of triangular prisms.It is constructed from a triangular tiling extruded into prisms....	(3⁶) 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....
J₄₇ A₉ G₁₆	t_0,2 x	Rhombi-trihexagonal prismatic Rhombitriangular-hexagonal prismatic honeycomb The rhombitriangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of hexagonal prisms, cubes, and triangular prisms in a ratio of 1:3:2....	(3.4.6.4) 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 ....
J₄₉ A₁₀ G₂₃	t_0,1,2 x	Omnitruncated trihexagonal prismatic Omnitruncated triangular-hexagonal prismatic honeycomb The omnitruncated triangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1:2:3....	(4.6.12) 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 ....
J₄₈ A₁₂ G₁₇	s x	Snub trihexagonal prismatic Snub triangular-hexagonal prismatic honeycomb The Snub triangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of hexagonal prisms and triangular prisms in a ratio of 1:8....	(3.3.3.3.6) 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 ....
J₆₅ A_11' G₁₃	:e x	elongated triangular prismatic Elongated triangular prismatic honeycomb The elongated triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of cubes and triangular prisms in a ratio of 1:2....	(3.3.3.4.4) 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 ....

Examples

All 28 of these tessellations are found in crystal

Crystal

A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions....
arrangements.

The alternated cubic honeycomb

Tetrahedral-octahedral honeycomb

The tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is comprised of alternating octahedron and tetrahedron in a ratio of 1:2....
is of special importance since its vertices form a cubic close-packing

Close-packing

In geometry, close-packing of spheres is the construction of an infinite regular arrangement of identical spheres so that they take up the greatest possible fraction of an infinite 3-dimensional space ....
of spheres. The space-filling truss

Truss

In architecture and structural engineering, a truss is a architectural structure comprising one or more triangular units constructed with straight slender members whose ends are connected at joints referred to as Vertex ....
of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell

Alexander Graham Bell

Alexander Graham Bell was an eminent scientist, Innovation and innovator who is credited with inventing the first practical telephone.Bell's father, grandfather, and brother had all been associated with work on elocution and speech, and both his mother and wife were deaf, profoundly influencing Bell's life's work....
and independently re-discovered by Buckminster Fuller

Buckminster Fuller

Richard Buckminster ?Bucky? Fuller was an American architect, author, designer, futurist, inventor, and visionary. He was the second president of Mensa International....
(who called it the octet truss and patented it in the 1940s). . Octet trusses are now among the most common types of truss used in construction.

Noncompact forms

If cells are allowed to be uniform tiling

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....
s, more uniform honeycombs can be defined:

Families:

C^~₂xA₁: [4,4]x[ ] Cubic prismatic slab honeycomb (3 forms)
H^~₂xA₁: [6,3]x[ ] Tri-hexagonal prismatic slab honeycomb (8 forms)
A^~₂xA₁: [Δ]x[ ] triangular prismatic slab (No new forms)
I^~₁xA₁xA₁: [∞]x[ ]x[ ] = Cubic column honeycomb (1 form)
I₂(p)xI^~₁: [p]x[∞] Prismatic column honeycomb
I^~₁xI^~₁xA₁: [∞]x[∞]x[ ] = [4,4]x[ ] - = (Same as cubic slab honeycomb family)

Examples (partially drawn): Cubic slab honeycomb and Alternated hexagonal slab honeycomb.

Hyperbolic forms

There are 9 Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
families of compact uniform honeycombs in hyperbolic 3-space

Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
, generated as Wythoff construction

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....
s, and represented by ring permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
s for each family.

From these 9 families, there are a total of 76 unique honeycombs generated:

[3,5,3] : # [5,3,4] : # [5,3,5] : # [5,3^1,1] : #[4,3,3,3:] : #[4,3,4,3:] : #[5,3,3,3:] : #[5,3,4,3:] : #[5,3,5,3:] :

The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-wythoffian exist. One known example is given with the family below.

[3,5,3] family

There are 9 forms, generated by ring permutations of the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
: [3,5,3] or One related non-wythoffian

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....
form is constructed from the vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps.

#	Honeycomb name Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction.... and Schläfli 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.... symbols	Cell counts/vertex and positions in honeycomb
#		0	1	2	3	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....	picture
1	icosahedral (Regular) t₀				(20) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
2	rectified icosahedral t₁	(2) (5.5.5) 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....			(3) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
3	truncated icosahedral t_0,1	(1) (5.5.5) 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....			(3) (4.6.6) Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
4	cantellated icosahedral t_0,2	(1) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	(2) (4.4.3) 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....		(2) (3.5.4.5)
5	Runcinated icosahedral t_0,3	(1) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	(5) (4.4.3) 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....	(5) (4.4.3) 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....	(1) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
6	bitruncated icosahedral t_1,2	(2) (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....			(2) (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....
7	cantitruncated icosahedral t_0,1,2	(1) (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....	(1) (4.4.3) 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....		(2) (4.6.10)
8	runcitruncated icosahedral t_0,1,3	(1) (3.5.4.5)	(1) (4.4.3) 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....	(2) (4.4.6) Hexagonal prism In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce...	(1) (4.6.6) Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
9	omnitruncated icosahedral t_0,1,2,3	(1) (4.6.10)	(1) (4.4.6) Hexagonal prism In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce...	(1) (4.4.6) Hexagonal prism In geometry, the hexagonal prism is a Prism with hexagonal base.It is an octahedron. However, the term octahedron is mainly used with "regular" in front or implied, hence not meaning a hexagonal prism; in the general meaning the term octahedron it is not much used because there are different types which have not much in common exce...	(1) (4.6.10)
[77]	partially truncated icosahedral pt	(4) (5.5.5) 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....			(12) (3.3.3.5) Pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps....

[5,3,4] family

There are 15 forms, generated by ring permutations of the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
: [5,3,4] or

#	Name of honeycomb Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Cells by location and count per vertex				Picture
#		0	1	2	3	Picture
10	order-4 dodecahedral (Regular)	-	-	-	(8) (5.5.5) 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....
11	Rectified order-4 dodecahedral	(2) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	-	-	(4) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
12	Rectified order-5 cubic	(5) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	-	-	(2) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
13	order-5 cubic (Regular)	(20) (4.4.4) 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....	-	-	-
14	Truncated order-4 dodecahedral	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	-	-	(4) (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....
15	Bitruncated order-5 cubic	(2) (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....	-	-	(2) (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....
16	Truncated order-5 cubic	(5) (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....	-	-	(1) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
17	Cantellated order-4 dodecahedral	(1) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	(2) (4.4.4) 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....	-	(2) (3.4.5.4) Rhombicosidodecahedron The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....
18	Cantellated order-5 cubic	(2) (3.4.4.4) Rhombicuboctahedron The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....	-	(2) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(1) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
19	Runcinated order-5 cubic	(1) (4.4.4) 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....	(3) (4.4.4) 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....	(3) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(1) (5.5.5) 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....
20	Cantitruncated order-4 dodecahedral	(1) (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....	(1) (4.4.4) 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....	-	(2) (4.6.10)
21	Cantitruncated order-5 cubic	(2) (4.6.8)	-	(1) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(1) (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....
22	Runcitruncated order-4 dodecahedral	(1) (3.4.4.4) Rhombicuboctahedron The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangle and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each....	(1) (4.4.4) 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....	(2) (4.4.10) Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (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....
23	Runcitruncated order-5 cubic	(1) (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....	(2) (4.4.8) Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(1) (3.4.5.4) Rhombicosidodecahedron The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....
24	Omnitruncated order-5 cubic	(1) (4.6.8)	(1) (4.4.8) Octagonal prism In geometry, the octagonal prism is the sixth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.4.10) Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.6.10)

[5,3,5] family

There are 9 forms, generated by ring permutations of the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
: [5,3,5] or

#	Name of honeycomb Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Cells by location and count per vertex
#		0	1	2	3
25	Order-5 dodecahedral t₀				(20) (5.5.5) 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....
26	rectified order-5 dodecahedral t₁	(2) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....			(5) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
27	truncated order-5 dodecahedral t_0,1	(1) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....			(5) (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....
28	cantellated order-5 dodecahedral t_0,2	(1) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	(2) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....		(2) (3.5.4.5)
29	Runcinated order-5 dodecahedral t_0,3	(1) (5.5.5) 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....	(3) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(3) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(1) (5.5.5) 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....
30	bitruncated order-5 dodecahedral t_1,2	(2) (4.6.6) Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....			(2) (4.6.6) Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
31	cantitruncated order-5 dodecahedral t_0,1,2	(1) (4.6.6) Truncated icosahedron The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....	(1) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....		(2) (4.6.10)
32	runcitruncated order-5 dodecahedral t_0,1,3	(1) (3.5.4.5)	(1) (4.4.5) Pentagonal prism In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....	(2) (4.4.10) Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (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....
33	omnitruncated order-5 dodecahedral t_0,1,2,3	(1) (4.6.10)	(1) (4.4.10) Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.4.10) Decagonal prism In geometry, the decagonal prism is the eighth in an infinite set of Prism formed by square sides and two regular polygon caps.If faces are all regular, it is a semiregular polyhedron....	(1) (4.6.10)

[5,3^1,1] family

There are 11 forms (4 of which are not seen above), generated by ring permutations of the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
: [5,3^1,1] or

#	Honeycomb name Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....	Cells by location (and count around each vertex)				Picture
#		0	1	0'	3	Picture
34	alternated order-5 cubic			(12) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	(20) (3.3.3) 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....
35	truncated alternated order-5 cubic	(1) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....		(2) (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....	(2) (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....
[11]	rectified order-4 dodecahedral (rectified alternated order-5 cubic)	(2) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....		(2) (3.5.3.5) Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	(2) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
[12]	rectified order-5 cubic (cantellated alternated order-5 cubic)	(1) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....		(1) (3.3.3.3.3) Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	(5) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
[15]	bitruncated order-5 cubic (cantitruncated alternated order-5 cubic)	(1) (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....		(1) (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....	(2) (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....
[14]	truncated order-4 dodecahedral (bicantellated alternated order-5 cubic)	(2) (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....		(2) (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....	(1) (3.3.3.3) Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
[10]	Order-4 dodecahedral (trirectified alternated order-5 cubic)	(4) (5.5.5) 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....		(4) (5.5.5) 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....
36	runcinated alternated order-5 cubic	(1) (3.3.3) 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....		(3) (3.4.4.4) Rhombicosidodecahedron The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....	(1) (3.3.3) 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....
[17]	cantellated order-4 dodecahedral (runcicantellated alternated order-5 cubic)	(1) (3.4.5.4)	(2) (4.4.4) 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....	(1) (3.4.5.4)	(1) (3.4.3.4) Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
37	runcitruncated alternated order-5 cubic	(1) (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....		(2) (4.6.10) Truncated icosidodecahedron The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....	(1) (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....
[20]	cantitruncated order-4 dodecahedral (omnitruncated alternated order-5 cubic)	(1) (4.6.10)	(1) (4.4.4) 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....	(1) (4.6.10)	(1) (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....