The 5-simplex, also commonly called the hexateron or hix, is the simplest possible non-degenerate polyteron. The full symmetry version has 6 regular pentachora as cells, joining 5 to a vertex, and is one of the 3 regular polytera. It is the 5-dimensional simplex.

5-simplex
Rank5
TypeRegular
Notation
Bowers style acronymHix
Coxeter diagramx3o3o3o3o ()
Schläfli symbol{3,3,3,3}
Tapertopic notation14
Elements
Tera6 pentachora
Cells15 tetrahedra
Faces20 triangles
Edges15
Vertices6
Vertex figurePentachoron, edge length 1
Petrie polygons60 skew hexagonal-triangular coils
Measures (edge length 1)
Circumradius
Edge radius
Face radius
Cell radius
Inradius
Hypervolume
Diteral angle
HeightsPoint atop pen:
 Dyad atop perp tet:
 Trig atop perp trig:
Central density1
Number of external pieces6
Level of complexity1
Related polytopes
ArmyHix
RegimentHix
Dual5-simplex
ConjugateNone
Abstract & topological properties
Flag count720
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
Flag orbits1
ConvexYes
NatureTame

It can be viewed as a segmentoteron in three ways: as a pentachoric pyramid, as a dyad atop perpendicular tetrahedron, and as a triangle atop perpendicular triangle. This makes it the triangular member of an infinite family of isogonal polygonal disphenoids.

Vertex coordinates Edit

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

  •  ,
  •  ,
  •  ,
  •  ,
  •  .

Another set of coordinates can be given in B3÷(B2×A1) symmetry as:

  •  ,
  •  ,
  •  .

This represents the polytope as the convex hull of three skew orthogonal dyads.

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

  •  .

Representations Edit

A regular hexateron has the following Coxeter diagrams:

  • x3o3o3o3o (         ) (full symmetry)
  • ox3oo3oo3oo&#x (A4 axial, pentachoric pyramid)
  • xo ox3oo3oo&#x (A3×A1 symmetry, tetrahedral scalene)
  • xo3oo ox3oo&#x (A2≀S2 axial, triangular disphenoid)
  • oxo3ooo3ooo&#x (A3 symmetry, tetrahedral pyramidal pyramid)
  • oxo oox3ooo&#x (A2×A1 symmetry, triangular scalenic pyramid)
  • xoo oxo oox&#x (B3 symmetry, digonal trisphenoid or digonal dihedral trigyroprism)
  • ooox ooxo&#x (K2 symmetry, digonal disphenoidal pyramidal pyramid)
  • ooox3oooo&#x (A2 symmetry, triangular symmetry only)
  • oooox&#x (A1 symmetry only)
  • oooooo&#x (no symmetry, fully irregular)

Variations Edit

The regular hexateron has 2 subsymmetrical forms that remain isogonal:

External links Edit