In geometry, **1 _{k2} polytope** is a uniform polytope in n-dimensions (n = k+4) constructed from the E

The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.

Each polytope is constructed from **1 _{k-1,2}** and (n-1)-demicube facets. Each has a vertex figure of a

The sequence ends with k=6 (n=10), as an infinite tessellation of 9-dimensional hyperbolic space.

The complete family of **1 _{k2} polytope** polytopes are:

- 5-cell:
**1**, (5 tetrahedral cells)_{02} -
**1**polytope, (16 5-cell, and 10 16-cell facets)_{12} -
**1**polytope, (54 demipenteract facets)_{22} -
**1**polytope, (56_{32}**1**and 126 demihexeract facets)_{22} -
**1**polytope, (240_{42}**1**and 2160 demihepteract facets)_{32} -
**1**honeycomb, tessellates Euclidean 8-space (∞_{52}**1**and ∞ demiocteract facets)_{42} -
**1**honeycomb, tessellates hyperbolic 9-space (∞_{62}**1**and ∞ demienneract facets)_{52}

n | 1_{k2} | Petrie polygon projection | Name Coxeter-Dynkin diagram | Facets | Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1_{k-1,2} | (n-1)-demicube | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | ||||

4 | 1_{02} | 1_{20} | -- | 5 1 _{10} | 5 | 10 | 10 | 5 | |||||

5 | 1_{12} | 1_{21} | 16 1 _{20} | 10 1 _{11} | 16 | 80 | 160 | 120 | 26 | ||||

6 | 1_{22} | 1_{22} | 27 1 _{12} | 27 1 _{21} | 72 | 720 | 2160 | 2160 | 702 | 54 | |||

7 | 1_{32} | 1_{32} | 56 1 _{22} | 126 1 _{31} | 576 | 10080 | 40320 | 50400 | 23688 | 4284 | 182 | ||

8 | 1_{42} | 1_{42} | 240 1 _{32} | 2160 1 _{41} | 17280 | 483840 | 2419200 | 3628800 | 2298240 | 725760 | 106080 | 2400 | |

9 | 1_{52} | 1_{52} (8-space tessellation) | ∞ 1 _{42} | ∞ 1 _{51} | ∞ | ||||||||

10 | 1_{62} | 1_{62} (9-space hyperbolic tessellation) | ∞ 1 _{52} | ∞ 1 _{61} | ∞ |

- k
_{21}polytope family - 2
_{k1}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.

**Alicia Boole Stott** was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Groningen. She is best known for introducing the term "polytope" for a convex solid in four dimensions, and having an impressive grasp of four-dimensional geometry from a very early age.

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, **demihypercubes** are a class of *n*-polytopes constructed from alternation of an *n*-hypercube, labeled as *hγ _{n}* for being

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 five-dimensional geometry, a **demipenteract** or **5-demicube** is a semiregular 5-polytope, constructed from a *5-hypercube* (penteract) with alternated vertices removed.

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 k_{21} polytope** is a polytope in

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

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-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.

- 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

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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