# Birectified 5-simplex

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Birectified 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymDot
Coxeter diagramo3o3x3o3o ()
Elements
Tera12 rectified pentachora
Cells30 tetrahedra, 30 octahedra
Faces120 triangles
Edges90
Vertices20
Vertex figureTriangular duoprism, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Inradius${\displaystyle {\frac {\sqrt {15}}{10}}\approx 0.38730}$
Hypervolume${\displaystyle {\frac {11{\sqrt {3}}}{80}}\approx 0.23816}$
Diteral anglesRap–oct–rap: ${\displaystyle \arccos \left(-{\frac {1}{5}}\right)\approx 101.53696^{\circ }}$
Rap–tet–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 complexity3
Related polytopes
ArmyDot
RegimentDot
DualBidodecateron
ConjugateNone
Abstract & topological properties
Flag count4320
Euler characteristic2
OrientableYes
Properties
SymmetryA5×2, order 1440
Flag orbits3
ConvexYes
NatureTame

The birectified 5-simplex, also called the dodecateron, is a convex noble uniform 5-polytope. It consists of 12 rectified pentachora as facets. 6 rectified pentachora join at each triangular duoprismatic vertex. As the name suggests, it is the birectification of the 5-simplex. It is the medial stage in the series of truncations between the regular 5-simplex and its dual.

It can be seen as a segmentoteron as rectified pentachoron atop inverted rectified pentachoron, or a rectified pentachoric alterprism.

## Vertex coordinates

The vertices of a birectified 5-simplex of edge length 1 are given by the following points:

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

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

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

## Representations

A birectified 5-simplex has the following Coxeter diagrams:

• o3o3x3o3o () (full symmetry)
• oo3xo3ox3oo&#x (A4 axial, rectified pentachoric alterprism)
• oxoo3ooxo oxoo3ooxo&#xt (A2×A2 axial, vertex-first)
• oxo xoo3oxooox&#xt (A3×A1 axial, tetrahedron-first)
• oxoo3xoxo3ooox&#xt (A3 symmetry)

## Related polytopes

The birectified 5-simplex is the colonel of a five-member regiment that also includes the biprismatododecateron, biprismatointercepted dodecateron, cellibiprismatohexateron, and cellidishexateron.