Uniform 1 k2 polytope

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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[edit]

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[edit]

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 4-simplex t0.svg 120
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.png
-- 5
110
3-simplex t0.svg
5 10 10
2-simplex t0.svg
5
3-simplex t0.svg
       
5 112 5-demicube.svg 121
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.png
16
120
4-simplex t0.svg
10
111
4-orthoplex.svg
16 80 160
2-simplex t0.svg
120
3-simplex t0.svg
26
4-simplex t0.svg4-orthoplex.svg
     
6 122 Up 1 22 t0 E6.svg 122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
27
112
5-demicube.svg
27
121
5-demicube.svg
72 720 2160
2-simplex t0.svg
2160
3-simplex t0.svg
702
4-simplex t0.svg4-orthoplex.svg
54
5-demicube.svg
   
7 132 Up2 1 32 t0 E7.svg 132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
56
122
Up 1 22 t0 E6.svg
126
131
6-demicube.svg
576 10080 40320
2-simplex t0.svg
50400
3-simplex t0.svg
23688
4-simplex t0.svg4-orthoplex.svg
4284
5-simplex t0.svg5-demicube.svg
182
Gosset 1 22 polytope.svg6-demicube.svg
 
8 142 Gosset 1 42 polytope petrie.svg 142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
240
132
Up2 1 32 t0 E7.svg
2160
141
7-demicube.svg
17280 483840 2419200
2-simplex t0.svg
3628800
3-simplex t0.svg
2298240
4-simplex t0.svg4-orthoplex.svg
725760
5-simplex t0.svg5-demicube.svg
106080
6-simplex t0.svg6-demicube.svgGosset 1 22 polytope.svg
2400
7-demicube.svg2 41 polytope petrie.svg
9 152 152
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
(8-space tessellation)

142
Gosset 1 42 polytope petrie.svg

151
8-demicube.svg
10 162 162
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
(9-space hyperbolic tessellation)

152

161
9-demicube.svg

See also[edit]

References[edit]

  • 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[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family / /
Uniform tiling {3[3]} δ3 3 3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 4 4
Uniform 5-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 6 6
Uniform 7-honeycomb {3[7]} δ7 7 7 222
Uniform 8-honeycomb {3[8]} δ8 8 8 133331
Uniform 9-honeycomb {3[9]} δ9 9 9 152251521
Uniform n-honeycomb {3[n]} δn n n 1k22k1k21