# Uniform 1 k2 polytope

In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 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 {3,3k,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 1k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a {31,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 1k2 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 1k2 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

## 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