Cantellated 5-simplex

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Cantellated 5-simplex
Rank5
TypeUniform
Notation
Bowers style acronymSarx
Coxeter diagramx3o3x3o3o ()
Elements
Tera15 tetrahedral prisms, 6 rectified pentachora, 6 small rhombated pentachora
Cells30 tetrahedra, 60 triangular prisms, 30 octahedra, 15 cuboctahedra
Faces20+60+120 triangles, 90 squares
Edges60+180
Vertices60
Vertex figureTriangular pyramidal prism, edge lengths 1 (base triangular prism and top edge) and 2 (sides)
Measures (edge length 1)
Circumradius
Hypervolume
Diteral anglesRap–tet–tepe:
 Srip–trip–tepe:
 Srip–oct–rap:
 Srip–co–srip:
Central density1
Number of external pieces27
Level of complexity16
Related polytopes
ArmySarx
RegimentSarx
DualTriangular-pyramidal tegmatic hexecontateron
ConjugateNone
Abstract & topological properties
Flag count11520
Euler characteristic2
OrientableYes
Properties
SymmetryA5, order 720
Flag orbits16
ConvexYes
NatureTame

The cantellated 5-simplex, also called the small rhombated hexateron or sarx, is a convex uniform 5-polytope. It consists of 15 tetrahedral prisms, 6 rectified pentachora, and 6 small rhombated pentachora. One rectified pentachoron, 2 tetrahedral prisms, and 3 small rhombated pentachora join at each vertex. As the name suggests, it is the cantellation of the 5-simplex.

Vertex coordinates[edit | edit source]

The vertices of a cantellated 5-simplex of edge length 1 can be given in 6 dimensions as all permutations of:

  • .

Representations[edit | edit source]

A cantellated 5-simplex has the following Coxeter diagrams:

  • x3o3x3o3o () (full symmetry)
  • oxx3xxo3oox3ooo&#xt (A4 axial, rectified pentachoron-first)
  • x(uo)xo x(ou)xx3o(xo)xo3o(oo)ox&#xt (A3×A1 axial, tetrahedral prism-first)
  • ox(ou)xx3oo(xo)xo xx(uo)xo3ox(ox)oo&#xt (A2×A2 axial, triangle-first)

Gallery[edit | edit source]

Related polytopes[edit | edit source]

The cantellated 5-simplex is the colonel of a regiment that includes 15 uniform members.

External links[edit | edit source]