# 6-simplex

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${\displaystyle {\frac {\sqrt {21}}{7}}\approx 0.65465}$
Edge radius${\displaystyle {\frac {\sqrt {35}}{14}}\approx 0.42258}$
Face radius${\displaystyle {\frac {\sqrt {42}}{21}}\approx 0.30861}$
Cell radius${\displaystyle {\frac {\sqrt {42}}{28}}\approx 0.23146}$
Teron radius${\displaystyle {\frac {\sqrt {35}}{35}}\approx 0.16903}$
Inradius${\displaystyle {\frac {\sqrt {21}}{42}}\approx 0.10911}$
Hypervolume${\displaystyle {\frac {\sqrt {7}}{5760}}\approx 0.00045933}$
Dipetal angle${\displaystyle \arccos \left({\frac {1}{6}}\right)\approx 80.40593^{\circ }}$
HeightsPoint atop hix: ${\displaystyle {\frac {\sqrt {21}}{6}}\approx 0.76376}$
Dyad atop perp pen: ${\displaystyle {\frac {\sqrt {35}}{10}}\approx 0.59161}$
Trig atop perp tet: ${\displaystyle {\frac {\sqrt {42}}{12}}\approx 0.54006}$
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.

## Vertex coordinates

The vertices of a regular heptapeton of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}}\right)}$,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}}\right)}$.

Much simpler coordinates can be given in seven dimensions, as all permutations of:

• ${\displaystyle \left({\frac {\sqrt {2}}{2}},\,0,\,0,\,0,\,0,\,0,\,0\right)}$.

## Representations

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)