Semiregular E-polytope

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In geometry, a semiregular E-polytope is a polytope in a dimensional family contained with an E_n Coxeter group, and containing only regular polytope facets.

Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polyopes, 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. The final (9th dimensional figure) is an infinite tessellation, which he called a 9-ic semiregular check.

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.

The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice.

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. (E3) triangular prism: -1₂₁ (2 Triangles and 3 square faces)
2. (E4) rectified 5-cell: 0₂₁, Tetroctahedric (5 tetrahedra and 5 octahedra cells)
3. (E5) demipenteract: 1₂₁, 5-ic semiregular figure (16 5-cell and 10 16-cell facets)
4. E6 polytope: 2₂₁, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets)
5. E7 polytope: 3₂₁, 7-ic semiregular figure (567 6-simplex and 126 6-orthoplex facets)
6. E8 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)

• A final form was not discovered by Gosset and is called the E9 lattice: 6₂₁. It is a tessellation in of hyperbolic 9-space constructed of (∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.

Each polytope is constructed from (n-1)-simplex and (n-1)-orthoplex facets, each has a vertex figure as the previous form. For example the rectified 5-cell has a vertex figure as a triangular prism.

The family is also named by Coxeter as k₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the k-node sequence.

[edit] Elements

Gosset semiregular figures

n-ic	k21	Graph	Name Coxeter-Dynkin diagram	Facets		Elements
				(n-1)-simplex	(n-1)-orthoplex	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces
3-ic	-121		Triangular prism	2 triangles	3 squares	6	9	5
4-ic	021		Rectified 5-cell	5 tetrahedron	5 octahedron	10	30	30	10
5-ic	121		Demipenteract	16 5-cell	10 16-cell	16	80	160	120	26
6-ic	221		E6 polytope	72 5-simplexes	27 5-orthoplexes	27	216	720	1080	648	99
7-ic	321		E7 polytope	576 6-simplexes	126 6-orthoplexes	56	756	4032	10080	12096	6048	702
8-ic	421		E8 polytope	17280 7-simplexes	2160 7-orthoplexes	240	6720	60480	241920	483840	483840	207360	19440
9-ic	521		E8 lattice	∞ 8-simplexes	∞ 8-orthoplexes	∞	∞	∞	∞	∞	∞	∞	∞
10-ic	621		E9 lattice	∞ 9-simplexes	∞ 9-orthoplexes	∞	∞	∞	∞	∞	∞	∞	∞

[edit] 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

[edit] External links

• PolyGloss v0.05: Gosset figures (Gossetoicosatope)
• Regular, SemiRegular, Regular faced and Archimedean polytopes