# Tetrahedral-octahedral honeycomb

Alternated cubic honeycomb

Type Uniform honeycomb
Family Alternated hypercubic honeycomb
Indexing[1] J21,31,51, A2
W9, G1
Schläfli symbol h{4,3,4}
{3[4]}
ht0,3{4,3,4}
h{4,4}h{∞}
Coxeter diagrams or
or

=
=
Cell types {3,3}, {3,4}
Face types triangle {3}
Edge figure [{3,3}.{3,4}]2
(rectangle)
Vertex figure

(cuboctahedron)
Symmetry group Fm3m (225)
Symmetry ½${\tilde{C}}_3$, [1+,4,3,4]
${\tilde{B}}_3$, [4,31,1]
${\tilde{A}}_3$×2, <[3[4]]>
Dual Dodecahedrille
rhombic dodecahedral honeycomb
Properties vertex-transitive, edge-transitive, face-transitive, quasiregular honeycomb

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb, half cubic honeycomb, half cubic cellulation, or tetragonal disphenoidal cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2.

It is vertex-transitive with 8 tetrahedra and 6 octahedra around each vertex. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge.

John Horton Conway calls this honeycomb a Tetroctahedrille, and its dual dodecahedrille.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

It is part of an infinite family of uniform tessellations called alternated hypercubic honeycombs, formed as an alternation of a hypercubic honeycomb and being composed of demihypercube and cross-polytope facets.

In this case of 3-space, the cubic honeycomb is alternated, reducing the cubic cells to tetrahedra, and the deleted vertices create octahedral voids. As such it can be represented by an extended Schläfli symbol h{4,3,4} as containing half the vertices of the {4,3,4} cubic honeycomb.

There's a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra.

## Cartesian coordinates

For an alternted cubic honeycomb, with edges parallel to the axes and with an edge length of 1, the Cartesian coordinates of the vertices are: (For all integral values: i,j,k with i+j+k even)

(i, j, k)
This diagram shows an exploded view of the cells surrounding each vertex.

## Symmetry

There are two reflective constructions and many alternated cubic honeycomb ones; examples:

Symmetry ${\tilde{B}}_3$, [4,31,1]
= ½${\tilde{C}}_3$, [1+,4,3,4]
${\tilde{A}}_3$, [3[4]]
= ½${\tilde{B}}_3$, [1+,4,31,1]
[[(4,3,4,2+)]] [(4,3,4,2+)]
Space group Fm3m (225) F43m (216) I43m (217) P43m (215)
Image
Types of tetrahedra 1 2 2 4
Coxeter
diagram
= = =

### Alternated cubic honeycomb slices

The alternated cubic honeycomb can be sliced into sections, where new square faces are created from inside of the octahedron. Each slice will contain up and downward facing square pyramids and tetrahedra sitting on their edges. A second slice direction needs no new faces and includes alternating tetrahedral and octahedral.

### Projection by folding

The alternated cubic honeycomb can be orthogonally projected into the planar square tiling by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the alternated cubic honeycomb creates two offset copies of the square tiling vertex arrangement of the plane:

Coxeter
group
${\tilde{A}}_3$ ${\tilde{C}}_2$
Coxeter
diagram
Image
Name alternated cubic honeycomb square tiling

## A3/D3 lattice

Its vertex arrangement represents an A3 lattice or D3 lattice.[2][3] It is the 3-dimensional case of a simplectic honeycomb. Its Voronoi cell is a rhombic dodecahedron, the dual of the cuboctahedron vertex figure for the tet-oct honeycomb.

The D+
3
packing can be constructed by the union of two D3 (or A3) lattices. The D+
n
packing is only a lattice for even dimensions. The kissing number is 22=4, (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[4]

+

The A*
3
or D*
3
lattice (also called A4
3
or D4
3
) can be constructed by the union of all four A3 lattices, and is identical to the vertex arrangement of the disphenoid tetrahedral honeycomb, dual honeycomb of the uniform bitruncated cubic honeycomb:[5] It is also the body centered cubic, the union of two cubic honeycombs in dual positions.

+ + + = dual of = + .

The kissing number of the D*
3
lattice is 8[6] and its Voronoi tessellation is a bitruncated cubic honeycomb, , containing all truncated octahedral Voronoi cells, .[7]

## Related honeycombs

The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.

Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Pm3m
(221)
4:2 [4,3,4] ×1 1, 2, 3, 4,
5, 6
Fm3m
(225)
2:2 [1+,4,3,4]
↔ [4,31,1]

Half 7, 11, 12, 13
I43m
(217)
4o:2 [[(4,3,4,2+)]] Half × 2 (7),
Fd3m
(227)
2+:2 [[1+,4,3,4,1+]]
↔ [[3[4]]]

Quarter × 2 10,
Im3m
(229)
8o:2 [[4,3,4]] ×2

(1), 8, 9

The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Fm3m
(225)
2:2 [4,31,1]
↔ [4,3,4,1+]

×1 1, 2, 3, 4
Fm3m
(225)
2:2 <[1+,4,31,1]>
↔ <[3[4]]>

×2 (1), (3)
Pm3m
(221)
4:2 <[4,31,1]> ×2

5, 6, 7, (6), 9, 10, 11

This honeycomb is one of five distinct uniform honeycombs[8] constructed by the ${\tilde{A}}_3$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Space
group
Fibrifold Square
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
F43m
(216)
1o:2 a1 [3[4]] ×1 (None)
Fd3m
(227)
2+:2 p2 [[3[4]]]
×2  3
Fm3m
(225)
2:2 d2 <[3[4]]>
↔ [4,3,31,1]

×2  1, 2
Pm3m
(221)
4:2 d4 [2[3[4]]]
↔ [4,3,4]

×4  4
Im3m
(229)
8o:2 r8 [4[3[4]]]
↔ [[4,3,4]]

×8  5,  (*)
Quasiregular polychora and honeycombs: h{4,p,q}
Space Finite Affine Compact Paracompact
Name h{4,3,3} h{4,3,4} h{4,3,5} h{4,3,6} h{4,4,3} h{4,4,4}
$\left\{3,{3\atop3}\right\}$ $\left\{3,{3\atop4}\right\}$ $\left\{3,{3\atop5}\right\}$ $\left\{3,{3\atop6}\right\}$ $\left\{4,{3\atop4}\right\}$ $\left\{4,{4\atop4}\right\}$
Coxeter
diagram
Image
Vertex
figure

r{p,3}

### Cantic cubic honeycomb

Cantic cubic honeycomb
Type Uniform honeycomb
Schläfli symbol h2{4,3,4}
Coxeter diagrams =
=
Cells t{3,4}
r{4,3}
t{3,3}
Vertex figure
Coxeter groups [4,31,1], ${\tilde{B}}_3$
[3[4]], ${\tilde{A}}_3$
Symmetry group Fm3m (225)
Dual half oblate octahedrille
Properties vertex-transitive

The cantic cubic honeycomb, cantic cubic cellulation or truncated half cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated octahedra, cuboctahedra and truncated tetrahedra in a ratio of 1:1:2. Its vertex figure is a rectangular pyramid.

John Horton Conway calls this honeycomb a truncated tetraoctahedrille, and its dual half oblate octahedrille.

#### Symmetry

It has two different uniform constructions. The ${\tilde{A}}_3$ construction can be seen with alternately colored truncated tetrahedra.

Symmetry [4,31,1], ${\tilde{B}}_3$
=<[3[4]]>
[3[4]], ${\tilde{A}}_3$
Space group Fm3m (225) F43m (216)
Coloring
Coxeter = =
Vertex figure

#### Related honeycombs

It is related to the cantellated cubic honeycomb. Rhombicuboctahedra are reduced to truncated octahedra, and cubes are reduced to truncated tetrahedra.

 cantellated cubic Cantic cubic , , rr{4,3}, r{4,3}, {4,3} , , t{3,4}, r{4,3}, t{3,3}

### Runcic cubic honeycomb

Runcic cubic honeycomb
Type Uniform honeycomb
Schläfli symbol h3{4,3,4}
Coxeter diagrams =
Face rr{4,3}
{4,3}
{3,3}
Vertex figure
Tapered triangular prism
Coxeter group ${\tilde{B}}_4$, [4,31,1]
Symmetry group Fm3m (225)
Dual quarter cubille
Properties vertex-transitive

The runcic cubic honeycomb or runcicantic cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2. Its vertex figure is a triangular prism, with a tetrahedron on one end, cube on the opposite end, and three rhombicuboctahedra around the trapezoidal sides.

John Horton Conway calls this honeycomb a 3-RCO-trille, and its dual quarter cubille.

### Related honeycombs

It is related to the runcinated cubic honycomb, with quarter of the cubes alternated into tetrahedra, and half expanded into rhombicuboctahedra.

 Runcinated cubic Runcic cubic = {4,3}, {4,3}, {4,3}, {4,3} , , , h{4,3}, rr{4,3}, {4,3} , ,

### Runcicantic cubic honeycomb

Runcicantic cubic honeycomb
Type Uniform honeycomb
Schläfli symbol h2,3{4,3,4}
Coxeter diagrams =
Coxeter group ${\tilde{B}}_4$, [4,31,1]
Vertex figure
Symmetry group Fm3m (225)
Dual half pyramidille
Properties vertex-transitive

The runcicantic cubic honeycomb or runcicantic cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cuboctahedra, truncated cubes and truncated tetrahedra in a ratio of 1:1:2. It is related to the runcicantellated cubic honeycomb.

John Horton Conway calls this honeycomb a f-tCO-trille, and its dual half pyramidille.

#### Related honeycombs

 Runcicantic cubic Runcicantellated cubic

### Gyrated tetrahedral-octahedral honeycomb

Gyrated tetrahedral-octahedral honeycomb
Type convex uniform honeycomb
Coxeter diagram

Schläfli symbol h{4,3,4}:g
h{6,3}h{∞}
s{3,6}h{∞}
s{3[3]}h{∞}
Cell types {3,3}, {3,4}
Vertex figure
Triangular orthobicupola G3.4.3.4
Space group P63/mmc (194)
[3,6,2+,∞]
Dual trapezo-rhombic dodecahedral honeycomb
Properties vertex-transitive, face-transitive

The gyrated tetrahedral-octahedral honeycomb or gyrated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of octahedra and tetrahedra in a ratio of 1:2.

It is vertex-uniform with 8 tetrahedra and 6 octahedra around each vertex.

It is not edge-uniform. All edges have 2 tetrahedra and 2 octahedra, but some are alternating, and some are paired.

It can be seen as reflective layers of this layer honeycomb:

#### Construction by gyration

This is a less symmetric version of another honeycomb, tetrahedral-octahedral honeycomb, in which each edge is surrounded by alternating tetrahedra and octahedra. Both can be considered as consisting of layers one cell thick, within which the two kinds of cell strictly alternate. Because the faces on the planes separating these layers form a regular pattern of triangles, adjacent layers can be placed so that each octahedron in one layer meets a tetrahedron in the next layer, or so that each cell meets a cell of its own kind (the layer boundary thus becomes a reflection plane). The latter form is called gyrated.

The vertex figure is called a triangular orthobicupola, compared to the tetrahedral-octahedral honeycomb whose vertex figure cuboctahedron in a lower symmetry is called a triangular gyrobicupola, so the gyro- prefix is reversed in usage.

Vertex figures
Honeycomb Gyrated tet-oct Reflective tet-oct
Image
Name triangular orthobicupola triangular gyrobicupola
Vertex figure
Symmetry D3h, order 12
D3d, order 12
(Oh, order 48)

#### Construction by alternation

Vertex figure with nonplanar 3.3.3.3 vertex configuration for the triangular bipyramids

The geometry can also be constructed with an alternation operation applied to a hexagonal prismatic honeycomb. The hexagonal prism cells become octahedra and the voids create triangular bipyramids which can be divided into pairs of tetrahedra of this honeycomb. This honeycomb with bipyramids is called a ditetrahedral-octahedral honeycomb. There are 3 Coxeter-Dynkin diagrams, which can be seen as 1, 2, or 3 colors of octahedra:

### Gyroelongated alternated cubic honeycomb

Gyroelongated alternated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol h{4,3,4}:ge
{3,6}h1{∞}
Coxeter diagram

Cell types {3,3}, {3,4}, (3.4.4)
Face types {3}, {4}
Vertex figure
Space group P63/mmc (194)
[3,6,2+,∞]
Properties vertex-uniform

The gyroelongated alternated cubic honeycomb or elongated triangular antiprismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2.

It is vertex-uniform with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex.

It is one of 28 convex uniform honeycombs.

The elongated alternated cubic honeycomb has the same arrangement of cells at each vertex, but the overall arrangement differs. In the elongated form, each prism meets a tetrahedron at one of its triangular faces and an octahedron at the other; in the gyroelongated form, the prism meets the same kind of deltahedron at each end.

### Elongated alternated cubic honeycomb

Elongated alternated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol h{4,3,4}:e
{3,6}g1{∞}
Cell types {3,3}, {3,4}, (3.4.4)
Vertex figure
triangular cupola joined to isosceles hexagonal pyramid
Space group  ?
Properties vertex-transitive

The elongated alternated cubic honeycomb or elongated triangular gyroprismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2.

It is vertex-uniform with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex. Each prism meets an octahedron at one end and a tetrahedron at the other.

It is one of 28 convex uniform honeycombs.

It has a gyrated form called the gyroelongated alternated cubic honeycomb with the same arrangement of cells at each vertex.

## Notes

1. ^ For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).
2. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html
3. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A3.html
4. ^ Conway (1998), p. 119
5. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds3.html
6. ^ Conway (1998), p. 120
7. ^ Conway (1998), p. 466
8. ^ [1], A000029 6-1 cases, skipping one with zero marks

## References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
• Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [2]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
• D. M. Y. Sommerville, An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.