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In geometry, a semiregular k21 polytope is a polytope in (k+4) dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by Coxeter as k21 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the k-node sequence.

Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the 5-ic semiregular figure.

Contents

Family members

The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of (∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.)

The family starts uniquely as 6-polytopes. The triangular prism and rectified 5-cell are included at the beginning for completeness. The demipenteract also exists in the demihypercube family.

They are also sometimes named by their symmetry group, like E6 polytope, although there are many uniform polytopes within the E6 symmetry.

The complete family of Gosset semiregular polytopes are:

  1. triangular prism: -121 (2 Triangles and 3 square faces)
  2. rectified 5-cell: 021, Tetroctahedric (5 tetrahedra and 5 octahedra cells)
  3. demipenteract: 121, 5-ic semiregular figure (16 5-cell and 10 16-cell facets)
  4. Gosset 2 21 polytope: 221, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets)
  5. Gosset 3 21 polytope: 321, 7-ic semiregular figure (567 6-simplex and 126 6-orthoplex facets)
  6. Gosset 4 21 polytope: 421, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets)
  7. E8 lattice: 521, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets)

Each polytope is constructed from (n-1)-simplex and (n-1)-orthoplex facets.

The orthoplex faces are constructed from the Coxeter group Dn-1 and have a Schlafli symbol of {31,n-1,1} rather than the regular {3n-2,4}. This construction is an implication of two "facet types". Half the facets around each orthoplex ridge are attached to another orthoplex, and the others are attached to a simplex. In contrast, every simplex ridge is attached to an orthoplex.

Each has a vertex figure as the previous form. For example the rectified 5-cell has a vertex figure as a triangular prism.

Elements

Gosset semiregular figures
n-ic k21 Graph Name
Coxeter-Dynkin
diagram
Facets Elements
(n-1)-simplex
{3n-2}
(n-1)-orthoplex
{3n-4,1,1}
Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces
3-ic -121 Triangular prism graphs.png Triangular prism
CDW dot.pngCDW 3b.pngCDW ring.pngCDW 2.pngCDW ring.png
2 triangles
Complete graph K3.svgTriangular prism simplex.png
CDW ring.pngCDW 3b.pngCDW dot.png
3 squares
2-orthoplex.svgTriangular prism orthoplex.png
CDW ring.pngCDW 2.pngCDW ring.png
6 9 5          
4-ic 021 E4 graph ortho.png Rectified 5-cell
CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-10.png
5 tetrahedron
Complete graph K4.svgUniform polyhedron-33-t0.png
CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
5 octahedron
3-orthoplex.svgUniform polyhedron-33-t1.png
CD downbranch-10.pngCD 3.pngCD dot.png
10 30 30 10        
5-ic 121 Demipenteract graph ortho.svg Demipenteract
CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD ring.png
16 5-cell
Complete graph K5.svgSchlegel wireframe 5-cell.png
CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
10 16-cell
4-orthoplex.svg Schlegel wireframe 16-cell.png
CD ring.pngCD 3.pngCD downbranch-00.pngCD 3.pngCD dot.png
16 80 160 120 26      
6-ic 221 E6 graph.svg 221 polytope
CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png
72 5-simplexes
Complete graph K6.svg
CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
27 5-orthoplexes
5-orthoplex.svg
CD ring.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
27 216 720 1080 648 99    
7-ic 321 E7 graph.svg 321 polytope
CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png
576 6-simplexes
Complete graph K7.svg
CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
126 6-orthoplexes
6-orthoplex.svg
CD ring.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
56 756 4032 10080 12096 6048 702  
8-ic 421 E8 graph.svg 421 polytope
CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png
17280 7-simplexes
Complete graph K8.svg
CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
2160 7-orthoplexes
7-orthoplex.svg
CD ring.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png
240 6720 60480 241920 483840 483840 207360 19440
9-ic
521
E8 lattice
CD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD ring.png
8-simplexes
Complete graph K9.svg
CDW ring.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.pngCDW 3b.pngCDW dot.png
8-orthoplexes
8-orthoplex.svg
CD ring.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD dot.pngCD 3b.pngCD downbranch-00.pngCD 3b.pngCD dot.png

See also

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • 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
  • G.Blind and R.Blind, "The semi-regular polyhedra", Commentari Mathematici Helvetici 66 (1991) 150--154
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 411-413: The Gosset Series: n21)

External links

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