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
Inradius
Hypervolume
Diteral anglesRap–oct–rap:
 Rap–tet–rap:
Height
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[edit | edit source]

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

  • ,
  • ,
  • ,
  • ,
  • ,
  • ,
  • .

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

  • .

Representations[edit | edit source]

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)

Gallery[edit | edit source]

Related polytopes[edit | edit source]

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

External links[edit | edit source]