Uniform_1_k2_polytope Knowpia

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 polytope polytopes are:

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

Elements

Gosset 1_k2 figures
n	1_k2	Petrie polygon projection	Name Coxeter-Dynkin diagram	Facets		Elements
n	1_k2	Petrie polygon projection	Name Coxeter-Dynkin diagram	1_k-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	∞

References

Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
- Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
- Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910.
- Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913.
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)

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	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
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
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

Summary

Family members

Elements

See also

References

External links