Uniform k ₂₁ polytope

In geometry, a uniform k₂₁ polytope is a polytope in k + 4 dimensions constructed from the E_n Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol 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.

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. 2 21 polytope: 2₂₁, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets)
5. 3 21 polytope: 3₂₁, 7-ic semiregular figure (576 6-simplex and 126 6-orthoplex facets)
6. 4 21 polytope: 4₂₁, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets)
7. 5 21 honeycomb: 5₂₁, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets)
8. 6 21 honeycomb: 6₂₁, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 9-orthoplex facets)

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

The orthoplex faces are constructed from the Coxeter group D_n−1 and have a Schläfli symbol of {3^1,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.

Gosset semiregular figures

n-ic	k21	Graph	Name Coxeter 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	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		221 polytope	72 5-simplexes	27 5-orthoplexes	27	216	720	1080	648	99
7-ic	321		321 polytope	576 6-simplexes	126 6-orthoplexes	56	756	4032	10080	12096	6048	702
8-ic	421		421 polytope	17280 7-simplexes	2160 7-orthoplexes	240	6720	60480	241920	483840	483840	207360	19440
9-ic	521		521 honeycomb	∞ 8-simplexes	∞ 8-orthoplexes	∞
10-ic	621		621 honeycomb	∞ 9-simplexes	∞ 9-orthoplexes	∞

• Uniform 2_k1 polytope family
• Uniform 1_k2 polytope family

• 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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 411–413: The Gosset Series: n₂₁)

• PolyGloss v0.05: Gosset figures (Gossetoicosatope)
• Regular, SemiRegular, Regular faced and Archimedean polytopes Archived 2011-07-19 at the Wayback Machine

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2	Uniform tiling	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21