SEMIREGULAR E-POLYTOPE

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 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:
# (E3) triangular prism: -121 (2 Triangles and 3 square faces)
# (E4) rectified 5-cell: 021, ''Tetroctahedric'' (5 tetrahedra and 5 octahedra cells)
# (E5) demipenteract: 121, ''5-ic semiregular figure'' (16 5-cell and 10 16-cell facets)
# E6 polytope: 221, ''6-ic semiregular figure'' (72 5-simplex and 27 5-orthoplex facets)
# E7 polytope: 321, ''7-ic semiregular figure'' (567 6-simplex and 126 6-orthoplex facets)
# E8 polytope: 421, ''8-ic semiregular figure'' (17280 7-simplex and 2160 7-orthoplex facets)
# E8 lattice: 521, ''9-ic semiregular check'' (∞ 8-simplex and ∞ 8-orthoplex facets)
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 longest node sequence.

Contents

Elements

References

External links

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 CDW dot.png CDW 3b.png CDW ring.png CDW 2.png CDW ring.png	2 triangles	3 squares	6	9	5
4-ic	021		Rectified 5-cell CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-10.png	5 tetrahedron	5 octahedron	10	30	30	10
5-ic	121		Demipenteract CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD ring.png	16 5-cell	10 16-cell	16	80	160	120	26
6-ic	221		E6 polytope CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD ring.png	72 5-simplexes	27 5-orthoplexes	27	216	720	1080	648	99
7-ic	321		E7 polytope CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD ring.png	576 6-simplexes	126 6-orthoplexes	56	756	4032	10080	12096	6048	702
8-ic	421		E8 polytope CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD ring.png	17280 7-simplexes	2160 7-orthoplexes	240	6720	60480	241920	483840	483840	206360	19440
9-ic	521		E8 lattice CD dot.png CD 3b.png CD dot.png CD 3b.png CD downbranch-00.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD dot.png CD 3b.png CD ring.png	∞ 8-simplexes	∞ 8-orthoplexes	∞	∞	∞	∞	∞	∞	∞	∞

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

External links

★ PolyGloss v0.05: Gosset figures (Gossetoicosatope)

★ Regular, SemiRegular, Regular faced and Archimedean polytopes