4 21 polytope
421 polytope | |
---|---|
Rank | 8 |
Type | Uniform |
Notation | |
Bowers style acronym | Fy |
Coxeter diagram | o3o3o3o3o3o3x *c3o () |
Elements | |
Zetta | 17280 7-simplices, 2160 7-orthoplexes |
Exa | 69120+138240 6-simplices |
Peta | 483840 5-simplices |
Tera | 483840 pentachora |
Cells | 241920 tetrahedra |
Faces | 60480 triangles |
Edges | 6720 |
Vertices | 240 |
Vertex figure | 321 polytope, edge length 1 |
Measures (edge length 1) | |
Circumradius | 1 |
Hypervolume | |
Dizettal angles | oca–hop-zee: |
zee–hop-zee: | |
Central density | 1 |
Number of external pieces | 19440 |
Level of complexity | 3 |
Related polytopes | |
Army | Fy |
Regiment | Fy |
Dual | Semistellatotriacontaditeri-icosiheptapeto-pentacontoctaexic diacositetracontazetton |
Conjugate | None |
Abstract & topological properties | |
Flag count | 2090188800 |
Euler characteristic | 0 |
Orientable | Yes |
Properties | |
Symmetry | E8, order 696729600 |
Convex | Yes |
Nature | Tame |
The 421 polytope (also called dischiliahecatonhexaconta-myriaheptachiliadiacosioctaconta-zetton, abbreviated as 2160-17280-zetton, fy, or E8 polytope) or the is a uniform 8-polytope. It consists of 2160 7-orthoplexes and 17280 7-simplices as facets. At each vertex 126 7-orthoplexes and 576 7-simplexes meet forming a 321 polytope as the vertex figure.
The 421 polytope contains the vertices and edges of the birectified 8-simplex, small exioctadecazetton, demiocteract, rectified 8-orthoplex, 321 polytope prism, triangular-221 polytope prism, pentachoric-rectified pentachoric duoprism, tetrahedral-demipenteractic duoprism, hexadecachoric duoprism, octahedral-triacontiditeric duoprism, square-hexacontatetrapetic duoprism, rectified octaexic prism, and triangular-hexateric duoprismatic prism.
Vertex coordinates[edit | edit source]
Coordinates for the vertices of a 421 polytope with edge length 1 are given by all permutations and sign changes of
- ,
as well as all even sign changes of
- .
Integral coordinates can be given in 9 dimensions, as all permutations of:
- ,
- ,
- .
These give an edge length , and come from the fact that it is the convex hull of an expanded 8-simplex and two opposite birectified 8-simplices.
Representations[edit | edit source]
A 421 polytope has the following Coxeter diagrams:
- o3o3o3o3o3o3x *c3o () (full symmetry)
- ooxoo3ooooo3ooooo3ooooo *c3ooooo3ooooo3oxoxo&#xt (E7 axial, vertex-first)
- oxooo3ooooo3oooxo *b3ooooo3ooooo3ooxoo3xooox&#xt (D7 axial, hecatonicosoctaexon-first)
- xo3oo3oo *b3oo3oo3oo3ox3oo&#zx (D8 subsymmetry)
- oxo3ooo3xoo3ooo3ooo3oox3ooo3oxo&#zx (A8 subsymmetry)
- xooxooo3oooooxo3ooxoooo3ooooooo3ooooxoo3oxooooo3oooxoox&#xt (A7 axial, octaexon-first)
- xoxooo3ooooxo3ooooox3xooxoo oxooox3ooxooo3oooxoo3oxooxo&#zx (A4×A4 subsymmetry)
- ooxoo3xoooo3oooxo *b3oooox ooxoo3oxooo3oooxo *f3oooox&#zx (D4×D4 subsymmetry)
- xoxo3xoox ooxo3oooo3oooo3oooo3ooox *e3oxoo&#zx (A6×A2 subsymmetry)
Gallery[edit | edit source]
-
E7 orthographic projection
-
E6
-
B2
-
B3
-
B4
-
B5
-
B6
-
B7
-
B8
-
A5
-
A7
External links[edit | edit source]
- Klitzing, Richard. "Fy".
- Wikipedia contributors. "421 polytope".