Truncated 24-cell honeycomb

Truncated 24-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t{3,4,3,3} tr{3,3,4,3} t2r{4,3,3,4} t2r{4,3,31,1} t{31,1,1,1}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
4-face type	Tesseract File:Schlegel wireframe 8-cell.png Truncated 24-cell File:Schlegel half-solid truncated 24-cell.png
Cell type	Cube Error creating thumbnail: Truncated octahedron Error creating thumbnail:
Face type	Square Triangle
Vertex figure	Error creating thumbnail: Tetrahedral pyramid
Coxeter groups	${\tilde{F}}_4$, [3,4,3,3] ${\tilde{B}}_4$, [4,3,31,1] ${\tilde{C}}_4$, [4,3,3,4] ${\tilde{D}}_4$, [31,1,1,1]
Properties	Vertex transitive

In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells.

It has a uniform alternation, called the snub 24-cell honeycomb. It is a snub from the ${\tilde{D}}_4$ construction. This truncated 24-cell has Schläfli symbol t{3^1,1,1,1}, and its snub is represented as s{3^1,1,1,1}.

Alternate names

• Truncated icositetrachoric tetracomb
• Truncated icositetrachoric honeycomb
• Cantitruncated 16-cell honeycomb
• Bicantitruncated tesseractic honeycomb

Symmetry constructions

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored truncated 24-cell facets. In all cases, four truncated 24-cells, and one tesseract meet at each vertex, but the vertex figures have different symmetry generators.

Coxeter group	Coxeter diagram	Facets	Vertex figure	Vertex figure symmetry (order)
${\tilde{F}}_4$ = [3,4,3,3]	Template:CDD	4: Template:CDD 1: Template:CDD	Error creating thumbnail:	Template:CDD, [3,3] (24)
${\tilde{F}}_4$ = [3,3,4,3]	Template:CDD	3: Template:CDD 1: Template:CDD 1: Template:CDD	Error creating thumbnail:	Template:CDD, [3] (6)
${\tilde{C}}_4$ = [4,3,3,4]	Template:CDD	2,2: Template:CDD 1: Template:CDD	File:Truncated 24-cell honeycomb C4 verf.png	Template:CDD, [2] (4)
${\tilde{B}}_4$ = [31,1,3,4]	Template:CDD	1,1: Template:CDD 2: Template:CDD 1: Template:CDD	Error creating thumbnail:	Template:CDD, [ ] (2)
${\tilde{D}}_4$ = [31,1,1,1]	Template:CDD	1,1,1,1: Template:CDD 1: Template:CDD	File:Truncated 24-cell honeycomb D4 verf.png	[ ]+ (1)

See also

Regular and uniform honeycombs in 4-space:

• Tesseractic honeycomb
• 16-cell honeycomb
• 24-cell honeycomb
• Rectified 24-cell honeycomb
• Snub 24-cell honeycomb
• 5-cell honeycomb
• Truncated 5-cell honeycomb
• Omnitruncated 5-cell honeycomb

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, Template:Isbn p. 296, Table II: Regular honeycombs
• 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, Template:Isbn [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 99
• Template:KlitzingPolytopes o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99

Template:Navbar-collapsible
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	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21