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 polytope 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	File:4-simplex t0.svg	120 Template:CDD	--	5 110 File:3-simplex t0.svg	5	10	10 File:2-simplex t0.svg	5 File:3-simplex t0.svg
5	112	File:5-demicube.svg	121 Template:CDD	16 120 File:4-simplex t0.svg	10 111 File:4-orthoplex.svg	16	80	160 File:2-simplex t0.svg	120 File:3-simplex t0.svg	26 File:4-simplex t0.svgFile:4-orthoplex.svg
6	122	File:Up 1 22 t0 E6.svg	122 Template:CDD	27 112 File:5-demicube.svg	27 121 File:5-demicube.svg	72	720	2160 File:2-simplex t0.svg	2160 File:3-simplex t0.svg	702 File:4-simplex t0.svgFile:4-orthoplex.svg	54 File:5-demicube.svg
7	132	File:Up2 1 32 t0 E7.svg	132 Template:CDD	56 122 File:Up 1 22 t0 E6.svg	126 131 File:6-demicube.svg	576	10080	40320 File:2-simplex t0.svg	50400 File:3-simplex t0.svg	23688 File:4-simplex t0.svgFile:4-orthoplex.svg	4284 File:5-simplex t0.svgFile:5-demicube.svg	182 File:Gosset 1 22 polytope.svgFile:6-demicube.svg
8	142	File:Gosset 1 42 polytope petrie.svg	142 Template:CDD	240 132 File:Up2 1 32 t0 E7.svg	2160 141 File:7-demicube.svg	17280	483840	2419200 File:2-simplex t0.svg	3628800 File:3-simplex t0.svg	2298240 File:4-simplex t0.svgFile:4-orthoplex.svg	725760 File:5-simplex t0.svgFile:5-demicube.svg	106080 File:6-simplex t0.svgFile:6-demicube.svgFile:Gosset 1 22 polytope.svg	2400 File:7-demicube.svgFile:2 41 polytope petrie.svg
9	152		152 Template:CDD (8-space tessellation)	∞ 142 File:Gosset 1 42 polytope petrie.svg	∞ 151 File:8-demicube.svg	∞
10	162		162 Template:CDD (9-space hyperbolic tessellation)	∞ 152	∞ 161 File:9-demicube.svg	∞

See also

• k₂₁ polytope family
• 2_k1 polytope family

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)