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Uniform k ₂₁ polytope

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Ingeometry, a uniform k₂₁ polytope is a polytopeink + 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.

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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:

triangular prism: −1₂₁ (2triangles and 3 square faces)
rectified 5-cell: 0₂₁, Tetroctahedric (5tetrahedra and 5 octahedra cells)
demipenteract: 1₂₁, 5-ic semiregular figure (165-cell and 10 16-cell facets)
2 21 polytope: 2₂₁, 6-ic semiregular figure (72 5-simplex and 27 5-orthoplex facets)
3 21 polytope: 3₂₁, 7-ic semiregular figure (576 6-simplex and 126 6-orthoplex facets)
4 21 polytope: 4₂₁, 8-ic semiregular figure (17280 7-simplex and 2160 7-orthoplex facets)
5 21 honeycomb: 5₂₁, 9-ic semiregular check tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets)
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.

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Gosset semiregular figures
n-ic	k₂₁	Graph	Name Coxeter diagram	Facets		Elements
n-ic	k₂₁	Graph	Name Coxeter diagram	(n − 1)-simplex {3ⁿ⁻²}	(n − 1)-orthoplex {3^n−4,1,1}	Vertices	Edges	Faces	Cells	4-faces	5-faces	6-faces	7-faces
3-ic	−1₂₁		Triangular prism	2triangles	3squares	6	9	5
4-ic	0₂₁		Rectified 5-cell	5tetrahedron	5octahedron	10	30	30	10
5-ic	1₂₁		Demipenteract	165-cell	1016-cell	16	80	160	120	26
6-ic	2₂₁		2₂₁ polytope	725-simplexes	275-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	∞

References

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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)

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v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	{3^[11]}	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

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Uniform k ₂₁ polytope

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Uniform k 21 polytope

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Uniform k ₂₁ polytope