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

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 has a vertex figure as an (n-1)-demicube, {3^1,n-2,1}.

The complete family of 2_k1 polytope polytopes are:

1. 5-cell: 2₀₁, (5 tetrahedra cells)
2. Pentacross: 2₁₁, (32 5-cell (2_0,1) facets)
3. Gosset 2 21 polytope: 2₂₁, (72 5-simplex and 27 5-orthoplex (2_1,1) facets)
4. Gosset 2 31 polytope: 2₃₁, (576 6-simplex and 56 2_2,1 facets)
5. Gosset 2 41 polytope: 2₄₁, (17280 7-simplex and 240 2_3,1 facets)
6. 5₂₁, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 2_4,1 facets)

Elements

Gosset 2_k1 figures

n	2k1	Petrie polygon projection	Name Coxeter-Dynkin diagram	Facets		Elements
				k21 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		Gosset 2_21 polytope {32,2,1}	27 {32,1,1}	72 {35}	27	216	720	1080	648	99
7	231		Gosset 2_31_polytope {32,3,1}	56 {32,2,1}	576 {36}	126	2016	10080	20160	16128	4788	632
8	241		Gosset 2_41_polytope {32,4,1}	240 {32,3,1}	17280 {37}	2160	69120	483840	1209600	1209600	544320	144960	17520 [+]
9	251		Gosset 2_51_lattice (8-space tessellation) {32,5,1}	∞ {32,3,1}	∞ {38}	∞	∞	∞	∞	∞	∞	∞	∞ [+]