# Hexateron

Hexateron
Rank5
TypeRegular
SpaceSpherical
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
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt{15}}{6} ≈ 0.64550}$
Edge radius${\displaystyle \frac{\sqrt6}{6} ≈ 0.40825}$
Face radius${\displaystyle \frac{\sqrt3}{6} ≈ 0.28868}$
Cell radius${\displaystyle \frac{\sqrt6}{12} ≈ 0.20412}$
Inradius${\displaystyle \frac{\sqrt{15}}{30} ≈ 0.12910}$
Hypervolume${\displaystyle \frac{\sqrt3}{480} ≈ 0.0036084}$
Diteral angle${\displaystyle \arccos\left(\frac15\right) ≈ 78.46301^\circ}$
HeightsPoint atop pen: ${\displaystyle \frac{\sqrt{15}}{5} ≈ 0.77460}$
Dyad atop perp tet: ${\displaystyle \frac{\sqrt6}{4} ≈ 0.61237}$
Trig atop perp trig: ${\displaystyle \frac{\sqrt3}{3} ≈ 0.57735}$
Central density1
Number of external pieces6
Level of complexity1
Related polytopes
ArmyHix
RegimentHix
DualHexateron
ConjugateNone
Abstract & topological properties
Flag count720
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
ConvexYes
NatureTame

The hexateron, hix or triangular disphenoid, also commonly called the 5-simplex, 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.

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:

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

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×A2 axial, triangular disphenoid)
• oxo3ooo3ooo&#x (A3 symmetry, tetrahedral pyramidal pyramid)
• oxo oox3ooo&#x A2×A1 symmetry, triangular scalene pyramid)
• xoo oxo oox&#x (A1×A1×A1 symmetry, digonal trisphenoid)
• ooox ooxo&#x (A1×A1 symmetry, disphenoidal pyramidal pyramid)
• ooox3oooo&#x (A2 symmetry, triangular symmetry only)
• oooox&#x (A1 symmetrry only)
• oooooo&#x (no symmetry, fully irregular)

## Variations

The regular hexateron has 2 subsymmetrical forms that remain isogonal:

## Related polytopes

o3o3o3o3o truncations
Name OBSA CD diagram Picture
Hexateron hix
Rectified hexateron rix
Dodecateron dot
Rectified hexateron rix
Hexateron hix
Truncated hexateron tix
Bitruncated hexateron bittix
Bitruncated hexateron bittix
Truncated hexateron tix
Small rhombated hexateron sarx
Small birhombidodecateron sibrid
Small rhombated hexateron sarx
Great rhombated hexateron garx
Great birhombidodecateron gibrid
Great rhombated hexateron garx
Small prismated hexateron spix
Small prismated hexateron spix
Prismatotruncated hexateron pattix
Prismatorhombated hexateron pirx
Prismatorhombated hexateron pirx
Prismatotruncated hexateron pattix
Great prismated hexateron gippix
Great prismated hexateron gippix
Celliprismated hexateron cappix
Cellirhombidodecateron card
Celligreatorhombated hexateron cograx
Celliprismated hexateron cappix
Celliprismatotruncatododecateron captid
Celligreatorhombated hexateron cograx