# Uniform k 21 polytope

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In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En 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.

## Contents

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.

## Family members

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 Dn1 and have a Schläfli symbol of {31,n1,1} rather than the regular {3n2,4}. 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.

## Elements

Gosset semiregular figures
n-ick21 Graph Name
Coxeter
diagram
Facets Elements
(n  1)-simplex
{3n2}
(n  1)-orthoplex
{3n4,1,1}
Vertices Edges Faces Cells 4-faces5-faces6-faces7-faces
3-ic121 Triangular prism
2 triangles

3 squares

695
4-ic021 Rectified 5-cell
5 tetrahedron

5 octahedron

10303010
5-ic121 Demipenteract
16 5-cell

10 16-cell

168016012026
6-ic221 221 polytope
72 5-simplexes

27 5-orthoplexes

27216720108064899
7-ic321 321 polytope
576 6-simplexes

126 6-orthoplexes

56756403210080120966048702
8-ic421 421 polytope
17280 7-simplexes

2160 7-orthoplexes

24067206048024192048384048384020736019440
9-ic521 521 honeycomb
8-simplexes

8-orthoplexes

10-ic621 621 honeycomb
9-simplexes

9-orthoplexes

## Related Research Articles

In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry, a five-dimensional polytope is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell.

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol {3,3,3k,1}.

In geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol {3,3k,2}.

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

## 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
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN   978-1-56881-220-5 (Chapter 26. pp. 411–413: The Gosset Series: n21)
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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 {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 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
E10Uniform 10-honeycomb{3[11]}δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21