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}.

Family members

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

The complete family of 2_k1 polytope polytopes are:

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

Elements

Gosset 2_k1 figures
n	2_k1	Petrie polygon projection	Name Coxeter-Dynkin diagram	Facets		Elements
n	2_k1	Petrie polygon projection	Name Coxeter-Dynkin diagram	2_k-1,1 polytope	(n-1)-simplex	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces
4	2₀₁		5-cell {3^2,0,1}	--	5 {3³}	5	10	10	5
5	2₁₁		pentacross {3^2,1,1}	16 {3^2,0,1}	16 {3⁴}	10	40	80	80	32
6	2₂₁		2 21 polytope {3^2,2,1}	27 {3^2,1,1}	72 {3⁵}	27	216	720	1080	648	99
7	2₃₁		2 31 polytope {3^2,3,1}	56 {3^2,2,1}	576 {3⁶}	126	2016	10080	20160	16128	4788	632
8	2₄₁		2 41 polytope {3^2,4,1}	240 {3^2,3,1}	17280 {3⁷}	2160	69120	483840	1209600	1209600	544320	144960	17520
9	2₅₁		2 51 honeycomb (8-space tessellation) {3^2,5,1}	∞ {3^2,4,1}	∞ {3⁸}	∞
10	2₆₁		2 61 honeycomb (9-space tessellation) {3^2,6,1}	∞ {3^2,5,1}	∞ {3⁹}	∞

References

External links

PolyGloss v0.05: Gosset figures (Gossetoctotope)

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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

Family members

Elements

See also

References

External links