In geometry, a semiregular k_{21} 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 Coxeter as k_{21} by its bifurcating CoxeterDynkin diagram, with a single ring on the end of the knode 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 5ic semiregular figure.
Contents 
The sequence as identified by Gosset ends as an infinite tessellation (spacefilling honeycomb) in 8space, 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 9space constructed of (∞ 9simplex and ∞ 9orthoplex facets with all vertices at infinity.)
The family starts uniquely as 6polytopes. The triangular prism and rectified 5cell 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:
Each polytope is constructed from (n1)simplex and (n1)orthoplex facets.
The orthoplex faces are constructed from the Coxeter group D_{n1} and have a Schlafli symbol of {3^{1,n1,1}} rather than the regular {3^{n2},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 5cell has a vertex figure as a triangular prism.
nic  k_{21}  Graph  Name CoxeterDynkin diagram 
Facets  Elements  

(n1)simplex {3^{n2}} 
(n1)orthoplex {3^{n4,1,1}} 
Vertices  Edges  Faces  Cells  4faces  5faces  6faces  7faces  
3ic  1_{21}  Triangular prism 
2 triangles 
3 squares 
6  9  5  
4ic  0_{21}  Rectified 5cell 
5 tetrahedron 
5 octahedron 
10  30  30  10  
5ic  1_{21}  Demipenteract 
16 5cell 
10 16cell 
16  80  160  120  26  
6ic  2_{21}  2_{21} polytope 
72 5simplexes 
27 5orthoplexes 
27  216  720  1080  648  99  
7ic  3_{21}  3_{21} polytope 
576 6simplexes 
126 6orthoplexes 
56  756  4032  10080  12096  6048  702  
8ic  4_{21}  4_{21} polytope 
17280 7simplexes 
2160 7orthoplexes 
240  6720  60480  241920  483840  483840  207360  19440  
9ic  5_{21} 
E8 lattice 
∞ 8simplexes 
∞ 8orthoplexes 
∞  ∞  ∞  ∞  ∞  ∞  ∞  ∞ 
Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
