Uniform 1 k2 polytope

In geometry, 1_k2 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 1_k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol {3,3^k,2}.

Family members

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

Each polytope is constructed from 1_k−1,2 and (n−1)-demicube facets. Each has a vertex figure of a {3^1,n−2,2} polytope, is a birectified n-simplex, t₂{3ⁿ}.

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

The complete family of 1_k2 polytopes are:

1. 5-cell: 1₀₂, (5 tetrahedral cells)
2. 1₁₂ polytope, (16 5-cell, and 10 16-cell facets)
3. 1₂₂ polytope, (54 demipenteract facets)
4. 1₃₂ polytope, (56 1₂₂ and 126 demihexeract facets)
5. 1₄₂ polytope, (240 1₃₂ and 2160 demihepteract facets)
6. 1₅₂ honeycomb, tessellates Euclidean 8-space (∞ 1₄₂ and ∞ demiocteract facets)
7. 1₆₂ honeycomb, tessellates hyperbolic 9-space (∞ 1₅₂ and ∞ demienneract facets)

Elements

Gosset 1_k2 figures

n	1k2	Petrie polygon projection	Name Coxeter-Dynkin diagram	Facets		Elements
				1k−1,2	(n−1)-demicube	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces
4	102		120	--	5 110	5	10	10	5
5	112		121	16 120	10 111	16	80	160	120	26
6	122		122	27 112	27 121	72	720	2160	2160	702	54
7	132		132	56 122	126 131	576	10080	40320	50400	23688	4284	182
8	142		142	240 132	2160 141	17280	483840	2419200	3628800	2298240	725760	106080	2400
9	152		152 (8-space tessellation)	∞ 142	∞ 151	∞
10	162		162 (9-space hyperbolic tessellation)	∞ 152	∞ 161	∞

See also

• k₂₁ polytope family
• 2_k1 polytope family

References

• 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

External links

• PolyGloss v0.05: Gosset figures (Gossetododecatope)

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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