Cantellated tesseractic honeycomb

Cantellated tesseractic honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t0,2{4,3,3,4} or rr{4,3,3,4} rr{4,3,31,1}
Coxeter-Dynkin diagram
4-face type	t0,2{4,3,3} t1{3,3,4} {3,4}×{}
Cell type	Octahedron Rhombicuboctahedron Triangular prism
Face type	{3}, {4}
Vertex figure	Cubic wedge
Coxeter group	${\tilde {C}}_{4}$ = [4,3,3,4] ${\tilde {B}}_{4}$ = [4,3,31,1]
Dual
Properties	vertex-transitive

In four-dimensional Euclidean geometry, the cantellated tesseractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 4-space. It is constructed by a cantellation of a tesseractic honeycomb creating cantellated tesseracts, and new 24-cell and octahedral prism facets at the original vertices.

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

References

• 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 [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p318 [2]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Klitzing, Richard. "4D Euclidean tesselations#4D". o3x3o *b3o4x, x4o3x3o4o - srittit - O90
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2	Uniform tiling	{3[3]}	δでるた3	hδでるた3	qδでるた3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δでるた4	hδでるた4	qδでるた4
E4	Uniform 4-honeycomb	{3[5]}	δでるた5	hδでるた5	qδでるた5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δでるた6	hδでるた6	qδでるた6
E6	Uniform 6-honeycomb	{3[7]}	δでるた7	hδでるた7	qδでるた7	222
E7	Uniform 7-honeycomb	{3[8]}	δでるた8	hδでるた8	qδでるた8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δでるた9	hδでるた9	qδでるた9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δでるた10	hδでるた10	qδでるた10
E10	Uniform 10-honeycomb	{3[11]}	δでるた11	hδでるた11	qδでるた11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δでるたn	hδでるたn	qδでるたn	1k2 • 2k1 • k21