Runcinated 16-cell honeycomb

Runcinated 16-cell honeycomb
(No image)
Type	Uniform 4-honeycomb
Schläfli symbol	t03{3,3,4,3}
Coxeter-Dynkin diagrams
4-face type	r{3,4,3} t03{4,3,3} r{3,4}x{} {3}x{3}
Cell type	r{3,4} {4,3} {3,3}
Face type	{3}, {4}
Vertex figure
Coxeter groups	${\displaystyle {\tilde {F}}_{4}}$, [3,4,3,3]
Properties	Vertex transitive

In four-dimensional Euclidean geometry, the runcinated 16-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a runcination of the regular 16-cell honeycomb, containing Rectified 24-cell, runcinated tesseract, cuboctahedral prism, and 3-3 duoprism cells.

Alternate names[]

• Runcinated hexadecachoric tetracomb/honeycomb
• Small prismated demitesseractic tetracomb (spaht)
• Small disicositetrachoric tetracomb

Related honeycombs[]

The [3,4,3,3], , Coxeter group generates 31 permutations of uniform tessellations, 28 are unique in this family and ten are shared in the [4,3,3,4] and [4,3,3^1,1] families. The alternation (13) is also repeated in other families.

F4 honeycombs
Extended symmetry	Extended diagram	Order	Honeycombs
[3,3,4,3]		×1	1, 3, 5, 6, , 9, , , 12
[3,4,3,3]		×1	2, 4, 7, 13, 14, 15, 16, 17, , , 20, , 23, , , 26, , ,
[(3,3)[3,3,4,3*]] =[(3,3)[31,1,1,1]] =[3,4,3,3]	= =	×4	(2), (4), (7), (13)

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 122
• Klitzing, Richard. "4D Euclidean tesselations". x3o3o4x3o - spaht - O122

v t Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21