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
4-face type	Tesseract Truncated 24-cell
Cell type	Cube Truncated octahedron
Face type	Square Triangle
Vertex figure	Tetrahedral pyramid
Coxeter groups	[math]\displaystyle{ {\tilde{F}}_4 }[/math], [3,4,3,3] [math]\displaystyle{ {\tilde{B}}_4 }[/math], [4,3,31,1] [math]\displaystyle{ {\tilde{C}}_4 }[/math], [4,3,3,4] [math]\displaystyle{ {\tilde{D}}_4 }[/math], [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 [math]\displaystyle{ {\tilde{D}}_4 }[/math] 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)
[math]\displaystyle{ {\tilde{F}}_4 }[/math] = [3,4,3,3]		4: 1:		, [3,3] (24)
[math]\displaystyle{ {\tilde{F}}_4 }[/math] = [3,3,4,3]		3: 1: 1:		, [3] (6)
[math]\displaystyle{ {\tilde{C}}_4 }[/math] = [4,3,3,4]		2,2: 1:		, [2] (4)
[math]\displaystyle{ {\tilde{B}}_4 }[/math] = [31,1,3,4]		1,1: 2: 1:		, [ ] (2)
[math]\displaystyle{ {\tilde{D}}_4 }[/math] = [31,1,1,1]		1,1,1,1: 1:		[ ]+ (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, ISBN 0-486-61480-8 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, 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]
• 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
• Klitzing, Richard. "4D Euclidean tesselations". https://bendwavy.org/klitzing/dimensions/flat.htm.  o4x3x3x4o, x3x3x *b3x4o, x3x3x *b3x *b3x, o3o3o4x3x, x3x3x4o3o - ticot - O99

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	[math]\displaystyle{ {\tilde{A}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{C}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{B}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{D}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
E2	Uniform tiling	{3[3]}	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	{3[4]}	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	{3[5]}	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	{3[6]}	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	{3[7]}	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	{3[8]}	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21