# Rectified 5-simplex

Rectified 5-simplex
Rank5
TypeUniform
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${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Hypervolume${\displaystyle {\frac {13{\sqrt {3}}}{240}}\approx 0.093819}$
Diteral anglesRap–tet–pen: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Rap–oct–rap: ${\displaystyle \arccos \left({\frac {1}{5}}\right)\approx 78.46304^{\circ }}$
Height${\displaystyle {\frac {\sqrt {15}}{5}}\approx 0.77460}$
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
Flag orbits4
ConvexYes
NatureTame

The rectified 5-simplex, also called the rectified hexateron or rix, is a convex uniform 5-polytope. 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 5-simplex.

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

## Vertex coordinates

The vertices of a rectified 5-simplex of edge length 1 are given by:

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

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

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

## Representations

A rectified 5-simplex 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 5-simplex 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 5-simplex is the colonel of a 7-member regiment. Its other members include the cellibiprismatointercepted hexateron, facetorectified hexateron, cellihexateron, biprismatodishexateron, cellintercepted dishexateron, and spinobiprismatohexateron.