Penticantellated 6-cube

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Penticantellated 6-cube
Rank6
TypeUniform
Notation
Bowers style acronymTopag
Coxeter diagramx4o3x3o3o3x ()
Elements
Peta240 square-tetrahedral duoprisms
192 small prismatodecachoric prisms
160 triangular-small rhombicuboctahedral duoprisms
60 small rhombated tesseractic prisms
64 runcinated 5-simplices
12 runcinated 5-cubes
Tera960+960 tetrahedral prisms
1280 triangular duoprisms
384 rectified pentachora
960+1920+1920 triangular-square duoprisms
960 octahedral prisms
384 small prismatodecachora
480 small rhombicuboctahedral prisms
120 small rhombated tesseracts
Cells960+1920 tetrahedrae
5×3840 triangular prisms
1920 octahedra
1440+5760 cubes
480 small rhombicuboctahedra
Faces3840+3840+7680 triangles, 960+5760+5760+11520 squares
Edges3840+5760+11520
Vertices3840
Vertex figureDiminished tetrahedral antifastegium, edge lengths 1 (base triangular prism and triangle of antifastegium facet) and 2 (remaining edges)
Measures (edge length 1)
Circumradius
Hypervolume
Dipetal anglesSpix–spid–spiddip:
 Spiddip–tepe–squatet:
 Trasirco–tisdip–squatet: 150°
 Srittip–sircope–trasirco:
 Spiddip–tisdip-trasirco:
 Sirn–srit–srittip, Spix–triddip–trasirco: 135°
 Spiddip–tisdip–srittip:
 Spix–ope–srittip:
Central density1
Number of external pieces728
Level of complexity176
Related polytopes
ArmyTopag
RegimentTopag
ConjugateNone
Abstract & topological properties
Flag count8524800
Euler characteristic0
OrientableYes
Properties
SymmetryB6, order 46080
ConvexYes
NatureTame

The penticantellated 6-cube, also called the penticantellated dodecapeton, teriprismated hexacontatetrapeton, or topag, is a convex uniform 6-polytope. It consists of 12 cantellated 5-cubes, 64 runcinated 5-simplices, 192 small prismatodecachoric prisms, 60 small rhombated tesseractic prisms, 240 square-tetrahedral duoprisms, and 160 triangular-small rhombicuboctahedral duoprisms. 1 cantellated 5-cube, 1 runcinated 5-simplex, 2 small prismatodecachoric prisms, 3 small rhombated tesseractic prisms, 1 square-tetrahedral duoprism, and 3 triangular-small rhombicuboctahedral duoprisms join at each vertex. As the name suggests, it is the penticantellation of the 6-cube.

The teriprismated hexacontatetrapeton contains the vertices of an inscribed steritruncated 5-cubic prism, octagonal-prismatorhombated hexadecachoric duoprism, and truncated cubic-great rhombicuboctahedral duoprism.

Vertex coordinates[edit | edit source]

The vertex coordinates of a penticantellated 6-cube, centered at the origin and with unit edge length, are given by all permutations of:

  • .

Gallery[edit | edit source]

Related polytopes[edit | edit source]

The regiment of the teriprismated hexacontatetrapeton contains 139 uniform members plus one fissary.

External links[edit | edit source]