Gosset 241 polytope

Orthographic projection
Blue vertex projections are singles, green vertex projections have multiplicity 3 and red one have multiplicity 6.
Type Uniform 8-polytope
Family 2k1 polytope
Schläfli symbol {32,4,1}
Coxeter-Dynkin diagram
7-faces 17520:
240 {32,3,1}
17280 {36}
6-faces 144960:
6720 {32,2,1}
138240 {35}
5-faces 544320:
60480 {32,1,1}
483840 {34}
4-faces 1209600:
241920 {32,0,1}
967680 {33}
Cells 1209600 {32}
Faces 483840 {3}
Edges 69120
Vertices 2160
Vertex figure demihepteract: {31,4,1}
Petrie polygon Octadecagon
Coxeter group E8, [34,2,1]
Properties convex

In 8-dimensional geometry, 241 is a uniform polytope, constructed from the E8 group. It is named by Coxeter as 241 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.

Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram.

Removing the node on the short branch leaves the 7-simplex:

Removing the node on the end of the 2-length branch leaves the 7-orthoplex in its alternated form: (411)

Every simplex facet touches an 7-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

Along with the semiregular polytope 421, discovered by Thorold Gosset in 1900, it is one of a family of 255 (28 − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram:

See also

References

Categories