# Rectified hexateron

Rectified hexateron
Rank5
TypeUniform
SpaceSpherical
Notation
Bowers style acronymRix
Coxeter diagramo3x3o3o3o ()
Elements
Tera6 pentachora, 6 rectified pentachora
Cells30 tetrahedra, 15 octahedra
Faces20+60 triangles
Edges60
Vertices15
Vertex figureTetrahedral prism, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt6}{3} ≈ 0.81650}$
Hypervolume${\displaystyle \frac{13\sqrt3}{240} ≈ 0.093819}$
Diteral anglesRap–tet–pen: ${\displaystyle \arccos\left(-\frac15\right) ≈ 101.53696^\circ}$
Rap–oct–rap: ${\displaystyle \arccos\left(\frac15\right) ≈ 78.46304^\circ}$
Height${\displaystyle \frac{\sqrt{15}}{5} ≈ 0.77460}$
Central density1
Number of external pieces12
Level of complexity4
Related polytopes
ArmyRix
RegimentRix
DualJoined hexateron
ConjugateNone
Abstract & topological properties
Flag count2880
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
ConvexYes
NatureTame

The rectified hexateron, or rix, also called the rectified 5-simplex, is a convex uniform polyteron. It consists of 6 regular pentachora and 6 rectified pentachora. Two pentachora and 4 rectified pentachora join at each tetrahedral prismatic vertex. As the name suggests, it is the rectification of the hexateron.

It is also a convex segmentoteron, as a pentachoron atop rectified pentachoron.

## Vertex coordinates

The vertices of a rectified hexateron of edge length 1 are given by:

• ${\displaystyle \left(-\frac{2\sqrt{15}}{15},\,-\frac{\sqrt{10}}{5},\,0,\,0,\,0\right),}$
• ${\displaystyle \left(-\frac{2\sqrt{15}}{15},\,\frac{\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),}$
• ${\displaystyle \left(-\frac{2\sqrt{15}}{15},\,\frac{\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle \left(-\frac{2\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,-\frac{\sqrt6}{4},\,0,\,0\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,-\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,-\frac{3\sqrt{10}}{20},\,\frac{\sqrt6}{12},\,\frac{\sqrt3}{6},\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,-\frac{\sqrt3}{3},\,0\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt6}{6},\,-\frac{\sqrt3}{6},\,±\frac12\right),}$
• ${\displaystyle \left(\frac{\sqrt{15}}{15},\,\frac{\sqrt{10}}{10},\,-\frac{\sqrt6}{6},\,\frac{\sqrt3}{6},\,±\frac12\right).}$

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

• ${\displaystyle \left(\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0,\,0,\,0,\,0\right).}$

## Representations

A rectified hexateron has the following Coxeter diagrams:

• o3x3o3o3o (full symmetry)
• xo3ox3oo3oo&#x (A4 axial, pentachoron atop rectified pentachoron)
• oxo oxo3oox3ooo&#xt (A3×A1 symmetry, vertex-first)
• x(oo)x3o(ox)o3o(oo)o&#xt (A3 symmetry, tetrahedron-first)
• oooo3ooxo3oxox&#xr (A3 symmetry)
• xoo3oxo oxo3oox&#xt (A2×A2 axial, triangle-first)
• oxoox ooxoo3oxoxo&#xr (A2×A1 symmetry)

## Related polytopes

A rectified hexateron can be diminished by removing tetrahedral prismatic pyramids. 3 such pyramids from non-adjacent vertices can be removed to form a scaliform tridiminished rectified hexateron.

The rectified hexateron is the colonel of a 7-member regiment. Its other members include the cellibiprismatointercepted hexateron, facetorectified hexateron, cellihexateron, biprismatodishexateron, cellintercepted dishexateron, and spinobiprismatohexateron.

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