In 8-dimensional geometry, the 2₅₁ honeycomb is a space-filling uniform tessellation. It is composed of 2₄₁ polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2_k1 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 short branch leaves the 8-simplex.

Removing the node on the end of the 5-length branch leaves the 2₄₁.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 1₅₁.

The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 0₅₁.

Related polytopes and honeycombs

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]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

v t e 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
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁