2 21 polytope
221 polytope | |
---|---|
Rank | 6 |
Type | Uniform |
Notation | |
Bowers style acronym | Jak |
Coxeter diagram | x3o3o3o3o *c3o () |
Elements | |
Peta | 72 5-simplices, 27 5-orthoplexes |
Tera | 216+432 pentachora |
Cells | 1080 tetrahedra |
Faces | 720 triangles |
Edges | 216 |
Vertices | 27 |
Vertex figure | Demipenteract, edge length 1 |
Measures (edge length 1) | |
Circumradius | |
Hypervolume | |
Dipetal angles | Tac–pen–hix: |
Tac–pen–tac: | |
Central density | 1 |
Number of external pieces | 99 |
Level of complexity | 3 |
Related polytopes | |
Army | Jak |
Regiment | Jak |
Dual | Semistellatotriacontaditeric icosiheptapeton |
Conjugate | None |
Abstract & topological properties | |
Flag count | 155520 |
Euler characteristic | 0 |
Orientable | Yes |
Properties | |
Symmetry | E6, order 51840 |
Convex | Yes |
Nature | Tame |
The 221 polytope (also called the icosiheptaheptacontadipeton or jak) is a convex uniform 6-polytope. It has 27 5-orthoplexes and 72 5-simplices as facets, with 10 5-orthoplexes and 16 5-simplices at each vertex forming a demipenteract as the vertex figure.
The 221 polytope contains the vertices of a hexateric prism, and is also the convex hull of 3 gyro-orthogonal triangular duoprisms.
It can tile 6-dimensional Euclidean space by itself, forming the 222 honeycomb.
Vertex coordinates[edit | edit source]
The vertices of a 221 polytope of edge length 1, centered at the origin, are given by:
- ,
- and all even sign changes of the first five coordinates,
- and all permutations of first 5 coordinates.
Representations[edit | edit source]
A 221 polytope has the following Coxeter diagrams:
- x3o3o3o3o *c3o () (full symmetry)
- oox3ooo3ooo3oxo *c3ooo&#xt (D5 axial, vertex-first)
- xox3ooo3ooo3oxo3ooo&#xt (A5 axial, hexateron-first)
- xoxo ooox3oxoo3oooo3ooxo&#xt (A4×A1 symmetry, edge-first)
- xo3oo3oo3ox3oo xo&#zx (A5×A1 axial)
- xoo3oxo oxo3oox oox3xoo&#zx (A2×A2×A2 symmetry, hull of 3 orthogonal triangular duoprisms)
Gallery[edit | edit source]
-
B6 orthographic projection
-
D5
-
D4, A2
-
A5
-
A4
-
A3, D3
Related polytopes[edit | edit source]
The 221 polytope is the colonel of a regiment with 9 uniform members, 5 fissary members, and one compound. Of its uniform members, three are noble.
It is the real analog of the Hessian polyhedron.
External links[edit | edit source]
- Bowers, Jonathan. "Category 1: Primary Polypeta" (#154).
- Klitzing, Richard. "jak".
- Wikipedia contributors. "2_21 polytope".