Runcicantitruncated tesseractic honeycomb

In four-dimensional Euclidean geometry, the runcicantitruncated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space.

Runcicantitruncated tesseractic honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t0,1,2,3{4,3,3,4}
Coxeter-Dynkin diagram
4-face type	Truncated 24-cell Truncated octahedral prism 4-8 duoprism Omnitruncated tesseract
Cell type	truncated cuboctahedron truncated octahedron octagonal prism hexagonal prism cube
Face type	{4}, {6}, {8}
Vertex figure	irr. 5-cell
Coxeter group	${\displaystyle {\tilde {C}}_{4}}$ = [4,3,3,4] ${\displaystyle {\tilde {B}}_{4}}$ = [4,3,31,1]
Dual
Properties	vertex-transitive

Related honeycombs

The [4,3,3,4], , Coxeter group generates 31 permutations of uniform tessellations, 21 with distinct symmetry and 20 with distinct geometry. The expanded tesseractic honeycomb (also known as the stericated tesseractic honeycomb) is geometrically identical to the tesseractic honeycomb. Three of the symmetric honeycombs are shared in the [3,4,3,3] family. Two alternations (13) and (17), and the quarter tesseractic (2) are repeated in other families.

C4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,3,4]:		×1	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
[[4,3,3,4]]		×2	(1), (2), (13), 18 (6), 19, 20
[(3,3)[1+,4,3,3,4,1+]] ↔ [(3,3)[31,1,1,1]] ↔ [3,4,3,3]	↔ ↔	×6	14, 15, 16, 17

The [4,3,3^1,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.

B4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[4,3,31,1]:		×1	5, 6, 7, 8
<[4,3,31,1]>: ↔[4,3,3,4]	↔	×2	9, 10, 11, 12, 13, 14, (10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]] ↔ [3[3,31,1,1]] ↔ [3,3,4,3]	↔ ↔	×3	1, 2, 3, 4
[(3,3)[1+,4,3,31,1]] ↔ [(3,3)[31,1,1,1]] ↔ [3,4,3,3]	↔ ↔	×12	20, 21, 22, 23

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• Demitesseractic honeycomb
• 24-cell honeycomb
• Truncated 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb