# Cantellated cubic honeycomb

Cantellated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol rr{4,3,4}
t0,2{4,3,4}
Coxeter-Dynkin diagram Template:CDD
Cells rr{4,3}
r{4,3}
{4,3}
Euler characteristic 0
Vertex figure
(Wedge)
Space group
Fibrifold notation
PmTemplate:Overlinem (221)
4:2
Coxeter group [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
Dual quarter oblate octahedrille
Properties vertex-transitive

The cantellated cubic honeycomb 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.

John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.

## Images

 It is closely related to the perovskite structure, shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb.

## Symmetry

There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells.

Vertex uniform colorings by cell
Construction Truncated cubic honeycomb Bicantellated alternate cubic
Coxeter group [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
=<[4,31,1]>
[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
Space group PmTemplate:Overlinem FmTemplate:Overlinem
Coxeter-Dynkin diagram Template:CDD Template:CDD
Coloring
Vertex figure
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1

## Related honeycombs

The [4,3,4], Template:CDD, 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
PmTemplate:Overlinem
(221)
4:2 [4,3,4] Template:CDD ×1 Template:CDD 1, Template:CDD 2, Template:CDD 3, Template:CDD 4,
Template:CDD 5, Template:CDD 6
FmTemplate:Overlinem
(225)
2:2 [1+,4,3,4]
= [4,31,1]
Template:CDD
= Template:CDD
Half Template:CDD 7, Template:CDD 11, Template:CDD 12, Template:CDD 13
ITemplate:Overline3m
(217)
4o:2 [[(4,3,4,2+)]] Template:CDD Half × 2 Template:CDD (7),
FdTemplate:Overlinem
(227)
2+:2 [[1+,4,3,4,1+]]
= [[3[4]]]
Template:CDD
= Template:CDD
Quarter × 2 Template:CDD 10,
ImTemplate:Overlinem
(229)
8o:2 [[4,3,4]] Template:CDD ×2

The [4,31,1], Template:CDD, 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
FmTemplate:Overlinem
(225)
2:2 [4,31,1]
= [4,3,4,1+]
Template:CDD
= Template:CDD
×1 Template:CDD 1, Template:CDD 2, Template:CDD 3, Template:CDD 4
FmTemplate:Overlinem
(225)
2:2 <[1+,4,31,1]>
= <[3[4]]>
Template:CDD
= Template:CDD
×2 Template:CDD (1), Template:CDD (3)
PmTemplate:Overlinem
(221)
4:2 <[4,31,1]> Template:CDD ×2