User:Tomruen/Gosset semiregular polytope
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Thorold Gosset's family of semiregular polytopes include:
- (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 is constructed from (n-1)-simplex and (n-1)-orthoplex facets, each has a vertex figure as the previous one.
The list ends as an infinite tessellation (space-filling honeycomb) in 8-space.
[edit] Elements
| n-ic | Graph | Names Symbol Coxeter-Dynkin |
Facets | Elements | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (n-1)-simplex | (n-1)-orthoplex | Vertices (0-faces) |
Edges (1-faces) |
Faces (2-faces) |
Cells (3-faces) |
(4-faces) | (5-faces) | (6-faces) | (7-faces) | |||
| 3-ic |  |
Triangular prism -121     |
2 triangles |
3 squares |
6 | 9 | 5 | |||||
| 4-ic |  |
Rectified 5-cell 021    |
5 tetrahedron |
5 octahedron |
10 | 30 | 30 | 10 | ||||
| 5-ic |  |
demipenteract 121   |
16 5-cell |
10 16-cell |
16 | 80 | 160 | 120 | 26 | |||
| 6-ic |  |
E6 polytope 221 |
72 5-simplexes |
27 5-orthoplexes |
27 | 216 | 720 | 1080 | 648 | 99 | ||
| 7-ic |  |
E7 polytope 321 |
576 6-simplexes |
126 6-orthoplexes |
56 | 756 | 4032 | 10080 | 12096 | 6048 | 702 | |
| 8-ic |  |
E8 polytope 421 |
17280 7-simplexes |
2160 7-orthoplexes |
240 | 6720 | 60480 | 241920 | 483840 | 483840 | 206360 | 19440 |
| 9-ic | E8 lattice 521 |
∞ 8-simplexes |
∞ 8-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
