1 52 honeycomb

152 honeycomb
(No image)
Type	Uniform tessellation
Family	1k2 polytope
Schläfli symbol	{3,35,2}
Coxeter symbol	152
Coxeter-Dynkin diagram	File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png
8-face types	142File:Gosset 1 42 polytope petrie.svg 151File:Demiocteract ortho petrie.svg
7-face types	132File:Gosset 1 32 petrie.svg 141File:Demihepteract ortho petrie.svg
6-face types	122File:Gosset 1 22 polytope.svg {31,3,1}File:Demihexeract ortho petrie.svg {35}File:6-simplex t0.svg
5-face types	121File:Demipenteract graph ortho.svg {34}File:5-simplex t0.svg
4-face type	111File:Cross graph 4.svg {33}File:4-simplex t0.svg
Cells	{32}File:3-simplex t0.svg
Faces	{3}File:2-simplex t0.svg
Vertex figure	birectified 8-simplex: t2{37} File:Birectified 8-simplex.png
Coxeter group	${\tilde{E}}_8$, [35,2,1]

In geometry, the 1₅₂ honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 1₄₂ and 1₅₁ facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1_k2 polytope family.

Construction

It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram.

File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

Removing the node on the end of the 2-length branch leaves the 8-demicube, 1₅₁.

File:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

Removing the node on the end of the 5-length branch leaves the 1₄₂.

File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 8-simplex, 0₅₂.

File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png

See also

• 5₂₁ honeycomb
• 2₅₁ honeycomb

References

• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Coxeter Regular Polytopes (1963), Macmillan Company
  • Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)
• 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] GoogleBook
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

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	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