Uniform 2 k1 polytope

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In geometry, 2k1 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 as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol {3,3,3k,1}.

Family members[edit]

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

Each polytope is constructed from (n-1)-simplex and 2k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demicube, {31,n-2,1}.

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

The complete family of 2k1 polytope polytopes are:

  1. 5-cell: 201, (5 tetrahedra cells)
  2. Pentacross: 211, (32 5-cell (201) facets)
  3. 221, (72 5-simplex and 27 5-orthoplex (211) facets)
  4. 231, (576 6-simplex and 56 221 facets)
  5. 241, (17280 7-simplex and 240 231 facets)
  6. 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)
  7. 261, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 251 facets)

Elements[edit]

Gosset 2k1 figures
n 2k1 Petrie
polygon

projection
Name
Coxeter-Dynkin
diagram
Facets Elements
2k-1,1 polytope (n-1)-simplex Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces
4 201 4-simplex t0.svg 5-cell
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.png
{32,0,1}
-- 5
{33}
3-simplex t0.svg
5 10 10
2-simplex t0.svg
5        
5 211 5-orthoplex.svg pentacross
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.png
{32,1,1}
16
{32,0,1}
4-simplex t0.svg
16
{34}
4-simplex t0.svg
10 40 80
2-simplex t0.svg
80
3-simplex t0.svg
32
4-simplex t0.svg
     
6 221 E6 graph.svg 2 21 polytope
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{32,2,1}
27
{32,1,1}
Cross graph 5.svg
72
{35}
5-simplex t0.svg
27 216 720
2-simplex t0.svg
1080
3-simplex t0.svg
648
4-simplex t0.svg
99
5-simplex t0.svgCross graph 5.svg
   
7 231 Gosset 2 31 polytope.svg 2 31 polytope
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{32,3,1}
56
{32,2,1}
E6 graph.svg
576
{36}
6-simplex t0.svg
126 2016 10080
2-simplex t0.svg
20160
3-simplex t0.svg
16128
4-simplex t0.svg
4788
5-simplex t0.svgCross graph 5.svg
632
6-simplex t0.svgE6 graph.svg
 
8 241 2 41 polytope petrie.svg 2 41 polytope
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
{32,4,1}
240
{32,3,1}
Gosset 2 31 polytope.svg
17280
{37}
7-simplex t0.svg
2160 69120 483840
2-simplex t0.svg
1209600
3-simplex t0.svg
1209600
4-simplex t0.svg
544320
5-simplex t0.svgCross graph 5.svg
144960
6-simplex t0.svgE6 graph.svg
17520
7-simplex t0.svgGosset 2 31 polytope.svg
9 251 2 51 honeycomb
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
(8-space tessellation)
{32,5,1}

{32,4,1}
2 41 polytope petrie.svg

{38}
8-simplex t0.svg
10 261 2 61 honeycomb
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.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 tessellation)
{32,6,1}

{32,5,1}

{39}
9-simplex t0.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]