4_{21} polytope  

Skew orthogonal projection inside Petrie polygon 

Type  Uniform 8polytope 
Family  k_{21} polytope 
Schläfli symbol  {3^{4,2,1}} 
Coxeter symbol  4_{21} 
CoxeterDynkin diagram  
7faces  19440 total: 2160 {3^{4,1,1}} 17280 {3^{6}} 
6faces  207360: 138240 {3^{4,0,1}} 68120 {3^{5}} 
5faces  483840 {3^{4}} 
4faces  483840 {3^{3}} 
Cells  241920 {3,3} 
Faces  60480 {3} 
Edges  6720 
Vertices  240 
Vertex figure  3_{21} polytope 
Petrie polygon  30gon 
Coxeter group  E_{8}, [3^{4,2,1}] 
Properties  convex 
In geometry, the Gosset 4_{21} polytope is an 8dimensional semiregular uniform polytope composed of 17,280 7simplex and 2,160 7orthoplex facets.
It was discovered by Thorold Gosset, who described it in his 1900 paper as an 8ic semiregular figure. It is the last finite semiregular figure in his enumeration, semiregular to him meaning that it contained only regular facets.
Donald Coxeter called it 4_{21} due to its bifurcating CoxeterDynkin diagram, having 3 branches of length 4, 2, and 1, and having a single ring on the final node of the 4 branch. As this graph is a representation of the simple Lie group E_{8}, the polytope is sometimes referred to as the E_{8} polytope.
For visualization this 8dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 240 vertices within a 30gonal regular polygon (called a Petrie polygon). Its 6720 edges are drawn between the 240 vertices. Specific higher elements (faces, cells, etc) can also be extracted and drawn on this projection.
Contents 
The facet information can be extracted from its CoxeterDynkin diagram.
Removing the node on the short branch leaves the 7simplex:
Removing the node on the end of the 2length branch leaves the 7orthoplex in its alternated form: (4_{11})
Every simplex facet touches an 7orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 3_{21} polytope.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8dimensional space.
The 240 vertices of the 4_{21} polytope can be constructed in two set, 112 (2^{2}C_{2}^{8}) with integer coordinates obtained from by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots (2^{7}) with halfinteger coordinates obtained from by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even).
The 4_{21} polytope can be projected into 3space as a physical vertexedge model as shown using Zome tools^{[1]} (Only a subset of edges are included) 
The E8 4_{21} polytope projected into 3space with all 6720 edges^{[2]} 
Projection into Petrie polygon for all unique vertex positions, 30fold symmetry 
Projection into Coxeter plane, 12fold symmetry. Red,orange,and yellow vertices have multiplicity of 1,8, and 24 
The 4_{21} polytope is last in a family called the k_{21} polytopes. The first polytope in this family is the semiregular triangular prism which is constructed from three squares (2orthoplexes) and two triangles (2simplexes).
As a 4dimensional regular complex polytope 3{3}3{3}3{3}3, Coxeter called the Witting polytope, after Alexander Witting.^{[3]}
It is one of a family of 255 (2^{8} − 1) convex uniform polytopes in eight dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this CoxeterDynkin diagram:
Among these 255, two other special polytopes in this family are: 1_{41} and 2_{41}, with a ring at the end of the other branches.
This polytope, along with the 8simplex, can create a uniform tessellaton of hyperbolic 8dimensional space, represented by symbol 5_{21} and CoxeterDynkin diagram:
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 
