1 52 honeycomb

class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2\|1₅₂ honeycomb
bgcolor=#ffffff align=center colspan=2\|(No image)
bgcolor=#e7dcc3\|Type	Uniform tessellation
bgcolor=#e7dcc3\|Family	1_k2 polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3^5,2}
bgcolor=#e7dcc3\|Coxeter symbol	1₅₂
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
bgcolor=#e7dcc3\|8-face types	1₄₂ 25px 1₅₁ 25px
bgcolor=#e7dcc3\|7-face types	1₃₂ 25px 1₄₁ 25px
bgcolor=#e7dcc3\|6-face types	1₂₂ 25px {3^1,3,1} 25px {3⁵} 25px
bgcolor=#e7dcc3\|5-face types	1₂₁ 25px {3⁴} 25px
bgcolor=#e7dcc3\|4-face type	1₁₁ 25px {3³} 25px
bgcolor=#e7dcc3\|Cells	{3²} 25px
bgcolor=#e7dcc3\|Faces	{3}25px
bgcolor=#e7dcc3\|Vertex figure	birectified 8-simplex: t₂{3⁷} 25px
bgcolor=#e7dcc3\|Coxeter group	${\tilde{E}}_8$ , [3^5,2,1]

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.

: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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

: {{CDD|nodea|3a|nodea|3a|nodea_1|3a|nodea|3a|nodea|3a|nodea|3a|nodea|3a|nodea}}

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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|978-0-471-01003-6}}, [https://books.google.com/books?id=fUm5Mwfx8rAC&dq=Coxeter&pg=PP1 GoogleBook]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

class="wikitable" align="right" style="margin-left:10px" width="280" !bgcolor=#e7dcc3 colspan=2\|1₅₂ honeycomb
bgcolor=#ffffff align=center colspan=2\|(No image)
bgcolor=#e7dcc3\|Type	Uniform tessellation
bgcolor=#e7dcc3\|Family	1_k2 polytope
bgcolor=#e7dcc3\|Schläfli symbol	{3,3^5,2}
bgcolor=#e7dcc3\|Coxeter symbol	1₅₂
bgcolor=#e7dcc3\|Coxeter-Dynkin diagram	{{CDD\|nodea\|3a\|nodea\|3a\|branch_01lr\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea\|3a\|nodea}}
bgcolor=#e7dcc3\|8-face types	1₄₂ 25px 1₅₁ 25px
bgcolor=#e7dcc3\|7-face types	1₃₂ 25px 1₄₁ 25px
bgcolor=#e7dcc3\|6-face types	1₂₂ 25px {3^1,3,1} 25px {3⁵} 25px
bgcolor=#e7dcc3\|5-face types	1₂₁ 25px {3⁴} 25px
bgcolor=#e7dcc3\|4-face type	1₁₁ 25px {3³} 25px
bgcolor=#e7dcc3\|Cells	{3²} 25px
bgcolor=#e7dcc3\|Faces	{3}25px
bgcolor=#e7dcc3\|Vertex figure	birectified 8-simplex: t₂{3⁷} 25px
bgcolor=#e7dcc3\|Coxeter group	${\tilde{E}}_8$ , [3^5,2,1]