# 5-simplex

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${\displaystyle {\frac {\sqrt {15}}{6}}\approx 0.64550}$
Edge radius${\displaystyle {\frac {\sqrt {6}}{6}}\approx 0.40825}$
Face radius${\displaystyle {\frac {\sqrt {3}}{6}}\approx 0.28868}$
Cell radius${\displaystyle {\frac {\sqrt {6}}{12}}\approx 0.20412}$
Inradius${\displaystyle {\frac {\sqrt {15}}{30}}\approx 0.12910}$
Hypervolume${\displaystyle {\frac {\sqrt {3}}{480}}\approx 0.0036084}$
Diteral angle${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46301^{\circ }}$
HeightsPoint atop pen: ${\displaystyle {\frac {\sqrt {15}}{5}}\approx 0.77460}$
Dyad atop perp tet: ${\displaystyle {\frac {\sqrt {6}}{4}}\approx 0.61237}$
Trig atop perp trig: ${\displaystyle {\frac {\sqrt {3}}{3}}\approx 0.57735}$
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

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

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

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

• ${\displaystyle \left(0,\,0,\,\pm {\frac {1}{2}},\,0,\,{\frac {\sqrt {6}}{6}}\right)}$ ,
• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,0,\,{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {6}}{12}}\right)}$ ,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,0,\,0,\,-{\frac {\sqrt {2}}{4}},\,-{\frac {\sqrt {6}}{12}}\right)}$ .

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:

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

## Representations

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

The regular hexateron has 2 subsymmetrical forms that remain isogonal: