Uniform 2 k1 polytope

In geometry, 2_k1 polytope is a uniform polytope in n dimensions (n = k + 4) constructed from the E_n Coxeter group. The family was named by their Coxeter symbol as 2_k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol {3,3,3^k,1}.

The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex (pentacross) in 5 dimensions, and the 4-simplex (5-cell) in 4 dimensions.

Each polytope is constructed from (n − 1)-simplex and 2_k−1,1 (n − 1)-polytope facets, each having a vertex figure as an (n − 1)-demicube, {3^1,n−2,1}.

The sequence ends with k = 6 (n = 10), as an infinite hyperbolic tessellation of 9-space.

The complete family of 2_k1 polytopes are:

1. 5-cell: 2₀₁, (5 tetrahedra cells)
2. Pentacross: 2₁₁, (32 5-cell (2₀₁) facets)
3. 2₂₁, (72 5-simplex and 27 5-orthoplex (2₁₁) facets)
4. 2₃₁, (576 6-simplex and 56 2₂₁ facets)
5. 2₄₁, (17280 7-simplex and 240 2₃₁ facets)
6. 2₅₁, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 2₄₁ facets)
7. 2₆₁, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 2₅₁ facets)

Gosset 2_k1 figures

n	2k1	Petrie polygon projection	Name Coxeter-Dynkin diagram	Facets		Elements
				2k−1,1 polytope	(n − 1)-simplex	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces
4	201		5-cell {32,0,1}	--	5 {33}	5	10	10	5
5	211		pentacross {32,1,1}	16 {32,0,1}	16 {34}	10	40	80	80	32
6	221		2 21 polytope {32,2,1}	27 {32,1,1}	72 {35}	27	216	720	1080	648	99
7	231		2 31 polytope {32,3,1}	56 {32,2,1}	576 {36}	126	2016	10080	20160	16128	4788	632
8	241		2 41 polytope {32,4,1}	240 {32,3,1}	17280 {37}	2160	69120	483840	1209600	1209600	544320	144960	17520
9	251		2 51 honeycomb (8-space tessellation) {32,5,1}	∞ {32,4,1}	∞ {38}	∞
10	261		2 61 honeycomb (9-space tessellation) {32,6,1}	∞ {32,5,1}	∞ {39}	∞

• k₂₁ polytope family
• 1_k2 polytope family

• A. Boole Stott (1910). "Geometrical deduction of semiregular from regular polytopes and space fillings" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. XI (1). Amsterdam: Johannes Müller. Archived from the original (PDF) on 29 April 2025.
• P. H. Schoute (1911). "Analytical treatment of the polytopes regularly derived from the regular polytopes" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Section I. XI (3). Amsterdam: Johannes Müller. Archived from the original (PDF) on 22 January 2025.
• P. H. Schoute (1913). "Analytical treatment of the polytopes regularly derived from the regular polytopes" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Sections II, III, IV. XI (5). Amsterdam: Johannes Müller. Archived from the original (PDF) on 22 February 2025.
• H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
• H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988

• PolyGloss v0.05: Gosset figures (Gossetoctotope)

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	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