6-simplex

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6-simplex
Rank6
TypeRegular
Notation
Bowers style acronymHop
Coxeter diagramx3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3}
Tapertopic notation15
Elements
Peta7 hexatera
Tera21 pentachora
Cells35 tetrahedra
Faces35 triangles
Edges21
Vertices7
Vertex figureHexateron, edge length 1
Petrie polygons360 heptagonal-heptagrammic-great heptagrammic coils
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Cell radius
Teron radius
Inradius
Hypervolume
Dipetal angle
HeightsPoint atop hix:
 Dyad atop perp pen:
 Trig atop perp tet:
Central density1
Number of external pieces7
Level of complexity1
Related polytopes
ArmyHop
RegimentHop
DualHeptapeton
ConjugateNone
Abstract & topological properties
Flag count5040
Euler characteristic0
OrientableYes
Properties
SymmetryA6, order 5040
Flag orbits1
ConvexYes
NatureTame

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]

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]