6simplex
6simplex  

Rank  6 
Type  Regular 
Notation  
Bowers style acronym  Hop 
Coxeter diagram  x3o3o3o3o3o () 
Schläfli symbol  {3,3,3,3,3} 
Tapertopic notation  1^{5} 
Elements  
Peta  7 5simplices 
Tera  21 pentachora 
Cells  35 tetrahedra 
Faces  35 triangles 
Edges  21 
Vertices  7 
Vertex figure  5simplex, edge length 1 
Petrie polygons  360 heptagonalheptagrammicgreat 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  A_{6}, order 5040 
Flag orbits  1 
Convex  Yes 
Nature  Tame 
The 6simplex (also called the heptapeton or hop) is the simplest possible nondegenerate 6polytope. 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 6polytope. It is the 6dimensional simplex. It is one of two uniform selfdual 6polytopes, the other being the great icosiheptapeton. It is also the 723 step prism and gyropeton, making it the simplest 6D step prism.
It can be obtained as a 6segmentotope in three ways: as a hexateric pyramid, dyad atop perpendicular pentachoron, or triangle atop perpendicular tetrahedron.
Gallery[edit  edit source]

A_{5} orthographic projection

A_{4}

A_{3}

A_{2}
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 (A_{5} axial, hexateric pyramid)
 xo ox3oo3oo3oo&#x (A_{4}×A_{1} axial, pentachric scalene)
 xo3oo ox3oo3oo&#x (A_{3}×A_{2} axial, tetrahedral tettene)
 oxo3ooo3ooo3ooo&#x (A_{4} only, pentachoric pyramidal pyramid)
 oxo oox3ooo3ooo&#xt (A_{3}×A_{1} axial, tetrahedral scalenic pyramid)
 oxo3ooo oox3ooo&#x (A_{2}×A_{2} axial, triangular disphenoidal pyramid)
 xoo oxo oox3ooo&#x (A_{1}×A_{2}×A_{1} axial, triangular scalenic scalene)
External links[edit  edit source]
 Bowers, Jonathan. "Category 1: Primary Polypeta" (#1).
 Klitzing, Richard. "hop".
 Wikipedia contributors. "6simplex".
 Hi.gher.Space Wiki Contributors. "Pyropeton".