6-simplex
6-simplex | |
---|---|
Rank | 6 |
Type | Regular |
Notation | |
Bowers style acronym | Hop |
Coxeter diagram | x3o3o3o3o3o () |
Schläfli symbol | {3,3,3,3,3} |
Tapertopic notation | 15 |
Elements | |
Peta | 7 hexatera |
Tera | 21 pentachora |
Cells | 35 tetrahedra |
Faces | 35 triangles |
Edges | 21 |
Vertices | 7 |
Vertex figure | Hexateron, edge length 1 |
Petrie polygons | 360 heptagonal-heptagrammic-great heptagrammic coils |
Measures (edge length 1) | |
Circumradius | |
Edge radius | |
Face radius | |
Cell radius | |
Teron radius | |
Inradius | |
Hypervolume | |
Dipetal angle | |
Heights | Point atop hix: |
Dyad atop perp pen: | |
Trig atop perp tet: | |
Central density | 1 |
Number of external pieces | 7 |
Level of complexity | 1 |
Related polytopes | |
Army | Hop |
Regiment | Hop |
Dual | Heptapeton |
Conjugate | None |
Abstract & topological properties | |
Flag count | 5040 |
Euler characteristic | 0 |
Orientable | Yes |
Properties | |
Symmetry | A6, order 5040 |
Convex | Yes |
Nature | Tame |
The 6-simplex (also called the heptapeton or hop) is the simplest possible non-degenerate 6-polytope. The full symmetry version has 7 regular hexatera as facets, joining 3 to a tetrahedron peak and 6 to a vertex, and is a regular 6-polytope. It is the 6-dimensional simplex. It is one of two uniform self-dual 6-polytopes, the other being the great icosiheptapeton. It is also the 7-2-3 step prism and gyropeton, making it the simplest 6D step prism.
It can be obtained as a 6-segmentotope in three ways: as a hexateric pyramid, dyad atop perpendicular pentachoron, or triangle atop perpendicular tetrahedron.
Gallery[edit | edit source]
-
A5 orthographic projection
-
A4
-
A3
-
A2
Vertex coordinates[edit | edit source]
The vertices of a regular heptapeton of edge length 1, centered at the origin, are given by:
- ,
- ,
- ,
- ,
- ,
- .
Much simpler coordinates can be given in seven dimensions, as all permutations of:
- .
Representations[edit | edit source]
A regular heptapeton has the following Coxeter diagrams:
- x3o3o3o3o3o () (full symmetry)
- ox3oo3oo3oo3oo&#x (A5 axial, hexateric pyramid)
- xo ox3oo3oo3oo&#x (A4×A1 axial, pentachric scalene)
- xo3oo ox3oo3oo&#x (A3×A2 axial, tetrahedral tettene)
- oxo3ooo3ooo3ooo&#x (A4 only, pentachoric pyramidal pyramid)
- oxo oox3ooo3ooo&#xt (A3×A1 axial, tetrahedral scalenic pyramid)
- oxo3ooo oox3ooo&#x (A2×A2 axial, triangular disphenoidal pyramid)
- xoo oxo oox3ooo&#x (A1×A2×A1 axial, triangular scalenic scalene)
External links[edit | edit source]
- Bowers, Jonathan. "Category 1: Primary Polypeta" (#1).
- Klitzing, Richard. "hop".
- Wikipedia contributors. "6-simplex".
- Hi.gher.Space Wiki Contributors. "Pyropeton".