In geometry, a **uniform k_{21} polytope** is a polytope in

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**: 6_{21}. 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 *E*_{6} symmetry.

The complete family of Gosset semiregular polytopes are:

- triangular prism: −1
_{21}(2 triangles and 3 square faces) - rectified 5-cell: 0
_{21},*Tetroctahedric*(5 tetrahedra and 5 octahedra cells) - demipenteract: 1
_{21},*5-ic semiregular figure*(16 5-cell and 10 16-cell facets) - 2 21 polytope: 2
_{21},*6-ic semiregular figure*(72 5-simplex and 27 5-orthoplex facets) - 3 21 polytope: 3
_{21},*7-ic semiregular figure*(576 6-simplex and 126 6-orthoplex facets) - 4 21 polytope: 4
_{21},*8-ic semiregular figure*(17280 7-simplex and 2160 7-orthoplex facets) - 5 21 honeycomb: 5
_{21},*9-ic semiregular check*tessellates Euclidean 8-space (∞ 8-simplex and ∞ 8-orthoplex facets) - 6 21 honeycomb: 6
_{21}, 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 {3^{n−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*.

n-ic | k_{21} | Graph | Name Coxeter diagram | Facets | Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(n − 1)-simplex {3 ^{n−2}} | (n − 1)-orthoplex {3 ^{n−4,1,1}} | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | ||||

3-ic | −1_{21} | Triangular prism | 2 triangles | 3 squares | 6 | 9 | 5 | ||||||

4-ic | 0_{21} | Rectified 5-cell | 5 tetrahedron | 5 octahedron | 10 | 30 | 30 | 10 | |||||

5-ic | 1_{21} | Demipenteract | 16 5-cell | 10 16-cell | 16 | 80 | 160 | 120 | 26 | ||||

6-ic | 2_{21} | 2_{21} polytope | 72 5-simplexes | 27 5-orthoplexes | 27 | 216 | 720 | 1080 | 648 | 99 | |||

7-ic | 3_{21} | 3_{21} polytope | 576 6-simplexes | 126 6-orthoplexes | 56 | 756 | 4032 | 10080 | 12096 | 6048 | 702 | ||

8-ic | 4_{21} | 4_{21} polytope | 17280 7-simplexes | 2160 7-orthoplexes | 240 | 6720 | 60480 | 241920 | 483840 | 483840 | 207360 | 19440 | |

9-ic | 5_{21} | 5_{21} honeycomb | ∞ 8-simplexes | ∞ 8-orthoplexes | ∞ | ||||||||

10-ic | 6_{21} | 6_{21} honeycomb | ∞ 9-simplexes | ∞ 9-orthoplexes | ∞ |

- Uniform 2
_{k1}polytope family - Uniform 1
_{k2}polytope family

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, **2 _{k1} polytope** is a uniform polytope in

In geometry, **1 _{k2} polytope** is a uniform polytope in n-dimensions constructed from the E

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

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

In geometry, the **5 _{21} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5

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 **E _{9} honeycomb** is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E

- 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: n_{21})

- PolyGloss v0.05: Gosset figures (Gossetoicosatope)
- Regular, SemiRegular, Regular faced and Archimedean polytopes

Space | Family | / / | ||||
---|---|---|---|---|---|---|

E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |

E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |

E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |

E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |

E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |

E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |

E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |

E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |

E^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |

E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |

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