Rectified 5-simplex

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

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

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

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

  • .

Representations[edit | edit source]

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)

Gallery[edit | edit source]

Related polytopes[edit | edit source]

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.

External links[edit | edit source]