# Runcitruncated cubic honeycomb

Runcitruncated cubic honeycomb

Type Uniform honeycomb
Schläfli symbol t0,1,3{4,3,4}
Coxeter-Dynkin diagrams Template:CDD
Vertex figure
(Trapezoidal pyramid)
Coxeter group [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
Space group
Fibrifold notation
PmTemplate:Overlinem (221)
4:2
Dual square quarter pyramidille
Properties vertex-transitive

The runcitruncated cubic honeycomb 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.

Its name is derived from its Coxeter-Dynkin diagram, Template:CDD 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.

## 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