The
rectified cubic honeycomb or
rectified cubic cellulation is a uniform space-filling
tessellation (or
honeycomb) in Euclidean 3-space. It is composed of
octahedra and
cuboctahedra in a ratio of 1:1, with a
square prism vertex figure.
John Horton Conway calls this honeycomb a
cuboctahedrille, and its dual an
oblate octahedrille. The
truncated cubic honeycomb or
truncated cubic cellulation is a uniform space-filling
tessellation (or
honeycomb) in Euclidean 3-space. It is composed of
truncated cubes and
octahedra in a ratio of 1:1, with an isosceles
square pyramid vertex figure.
John Horton Conway calls this honeycomb a
truncated cubille, and its dual
pyramidille. The
bitruncated cubic honeycomb is a space-filling
tessellation (or
honeycomb) in
Euclidean 3-space made up of
truncated octahedra (or, equivalently,
bitruncated cubes). It has four
truncated octahedra around each vertex, in a
tetragonal disphenoid vertex figure. Being composed entirely of
truncated octahedra, it is
cell-transitive. It is also
edge-transitive, with 2 hexagons and one square on each edge, and
vertex-transitive. It is one of 28
uniform honeycombs.
John Horton Conway calls this honeycomb a
truncated octahedrille in his
Architectonic and catoptric tessellation list, with its dual called an
oblate tetrahedrille, also called a
disphenoid tetrahedral honeycomb. Although a regular
tetrahedron can not tessellate space alone, this dual has identical
disphenoid tetrahedron cells with
isosceles triangle faces. The
alternated bitruncated cubic honeycomb or
bisnub cubic honeycomb is non-uniform, with the highest symmetry construction reflecting an alternation of the uniform bitruncated cubic honeycomb. A lower-symmetry construction involves regular icosahedra paired with golden icosahedra (with 8 equilateral triangles paired with 12 golden triangles). There are three constructions from three related
Coxeter diagrams: , , and . These have symmetry [4,3+,4], [4,(31,1)+] and [3[4+ respectively. The first and last symmetry can be doubled as
4,3+,4 and
3[4]+. This honeycomb is represented in the boron atoms of the
α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice. The
cantellated cubic honeycomb or
cantellated cubic cellulation is a uniform space-filling
tessellation (or
honeycomb) in Euclidean 3-space. It is composed of
rhombicuboctahedra,
cuboctahedra, and
cubes in a ratio of 1:1:3, with a
wedge vertex figure.
John Horton Conway calls this honeycomb a
2-RCO-trille, and its dual
quarter oblate octahedrille. The
cantitruncated cubic honeycomb or
cantitruncated cubic cellulation is a uniform space-filling
tessellation (or
honeycomb) in Euclidean 3-space, made up of
truncated cuboctahedra,
truncated octahedra, and
cubes in a ratio of 1:1:3, with a
mirrored sphenoid vertex figure.
John Horton Conway calls this honeycomb a
n-tCO-trille, and its dual
triangular pyramidille. Its dual of the
cantitruncated cubic honeycomb is called a
triangular pyramidille, with
Coxeter diagram, . These honeycomb cells represent the fundamental domains of {\tilde{B}}_3 symmetry. A cell can be as 1/24 of a translational cube with vertices positioned: taking two corners, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge. : The
alternated cantitruncated cubic honeycomb or
snub rectified cubic honeycomb contains three types of cells:
snub cubes,
icosahedra (with
Th symmetry),
tetrahedra (as tetragonal disphenoids), and new tetrahedral cells created at the gaps.Although it is not uniform, constructionally it can be given as
Coxeter diagrams or . Despite being non-uniform, there is a near-miss version with two edge lengths shown below, one of which is around 4.3% greater than the other. The snub cubes in this case are uniform, but the rest of the cells are not. The
cantic snub cubic honeycomb is constructed by snubbing the
truncated octahedra in a way that leaves only
rectangles from the
cubes (square prisms). It is not uniform but it can be represented as
Coxeter diagram . It has
rhombicuboctahedra (with
Th symmetry),
icosahedra (with
Th symmetry), and
triangular prisms (as
C2v-symmetry wedges) filling the gaps. The
runcitruncated cubic honeycomb or
runcitruncated cubic cellulation is a uniform
space-filling tessellation (or
honeycomb) in Euclidean 3-space. It is composed of
rhombicuboctahedra,
truncated cubes,
octagonal prisms, and
cubes in a ratio of 1:1:3:3, with an
isosceles-trapezoidal pyramid vertex figure. Its name is derived from its
Coxeter diagram, with three ringed nodes representing 3 active mirrors in the
Wythoff construction from its relation to the
regular cubic honeycomb.
John Horton Conway calls this honeycomb a
1-RCO-trille, and its dual
square quarter pyramidille. Its dual is
square quarter pyramidille, with
Coxeter diagram . Faces exist in 3 of 4 hyperplanes of the [4,3,4], {\tilde{C}}_3 Coxeter group. Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center. : An
alternated omnitruncated cubic honeycomb or
omnisnub cubic honeycomb can be constructed by
alternation of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given
Coxeter diagram: and has symmetry
4,3,4+. It makes
snub cubes from the
truncated cuboctahedra,
square antiprisms from the
octagonal prisms, and creates new
tetrahedral cells from the gaps. Its dual is a space-filling honeycomb constructed as the dual of the
alternated omnitruncated cubic honeycomb. The 24 cells fit around a vertex, making a chiral
octahedral symmetry that can be stacked in all 3-dimensions: : The
runcic cantitruncated cubic honeycomb or
runcic cantitruncated cubic cellulation is constructed by removing alternating long rectangles from the octagons and is not uniform, but it can be represented as
Coxeter diagram . It has
rhombicuboctahedra (with
Th symmetry),
snub cubes, two kinds of
cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a
cube but with
D2d symmetry), and
triangular prisms (as
C2v-symmetry wedges) filling the gaps. The
biorthosnub cubic honeycomb is constructed by removing alternating long rectangles from the octagons orthogonally and is not uniform, but it can be represented as
Coxeter diagram . It has
rhombicuboctahedra (with
Th symmetry) and two kinds of
cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a
cube but with
D2d symmetry). The
truncated square prismatic honeycomb or
tomo-square prismatic cellulation is a space-filling
tessellation (or
honeycomb) in
Euclidean 3-space. It is composed of
octagonal prisms and
cubes in a ratio of 1:1. It is constructed from a
truncated square tiling extruded into prisms. It is one of 28
convex uniform honeycombs. The
snub square prismatic honeycomb or
simo-square prismatic cellulation is a space-filling
tessellation (or
honeycomb) in
Euclidean 3-space. It is composed of
cubes and
triangular prisms in a ratio of 1:2. It is constructed from a
snub square tiling extruded into prisms. It is one of 28
convex uniform honeycombs. A
snub square antiprismatic honeycomb can be constructed by
alternation of the truncated square prismatic honeycomb, although it can not be made uniform, but it can be given
Coxeter diagram: and has symmetry [4,4,2,∞]+. It makes
square antiprisms from the
octagonal prisms,
tetrahedra (as tetragonal disphenoids) from the
cubes, and two tetrahedra from the
triangular bipyramids. ==See also ==