1 52 honeycomb

152 honeycomb
(No image)
Type	Uniform tessellation
Family	1k2 polytope
Schläfli symbol	{3,35,2}
Coxeter symbol	152
Coxeter-Dynkin diagram
8-face types	142 151
7-face types	132 141
6-face types	122 {31,3,1}25px {35}
5-face types	121 {34}
4-face type	111 {33}
Cells	{32}
Faces	{3}
Vertex figure	birectified 8-simplex: t2{37}
Coxeter group	[math]\displaystyle{ {\tilde{E}}_8 }[/math], [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.

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

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

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

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	[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