Runcinated 6-cube

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Runcinated 6-cube
Rank6
TypeUniform
Notation
Bowers style acronymSpox
Coxeter diagramx4o3o3x3o3o ()
Elements
Peta240 square-tetrahedral duoprisms
192 rectified pentachoric prisms
64 birectified 5-simplices
12 runcinated 5-cubes
Tera960 tetrahedral prisms
384+384 rectified pentachora
960 triangular-square duoprisms
960 octahedral prisms
60 small disprismatotesseractihexadecachora
Cells960+960 tetrahedra
1920+3840 triangular prisms
1920 octahedra
160+1440 cubes
Faces3840+3840 triangles, 960+5760 squares
Edges1920+5760
Vertices1280
Vertex figureTriangle atop gyrated triangular duoprism, edge lengths 1 (base triangular duoprism), 2 (top triangle and side edges)
Measures (edge length 1)
Circumradius
Hypervolume
Dipetal anglesDot–rap–rappip:
 Rappip–tepe–squatet:
 Span–tisdip–squatet: 120°
 Span–ope–rappip:
 Span–rap–dot:
 Span–sidpith–span: 90°
Central density1
Number of external pieces508
Level of complexity56
Related polytopes
ArmySpox
RegimentSpox
ConjugateQuasiprismated hexeract
Abstract & topological properties
Flag count2580480
Euler characteristic0
OrientableYes
Properties
SymmetryB6, order 46080
ConvexYes
NatureTame

The runcinated 6-cube, also called the runcinated dodecapeton, small prismated hexeract, or spox, is a convex uniform 6-polytope. It consists of 12 runcinated 5-cubes, 64 birectified 5-simplices, 192 rectified pentachoric prisms, and 240 square-tetrahedral duoprisms. 3 runcinated 5-cubes, 1 birectified 5-simplex, 3 rectified pentachoric prisms, and 3 square-tetrahedral duoprisms join at each vertex. As the name suggests, it is the runcination of the 6-cube.

The runcinated 6-cube contains the vertices and edges of the small rhombated 5-cubic prism, square-truncated tesseractic duoprism, octagonal-small rhombated tesseractic duoprism, truncated cubic-small rhombicuboctahedral duoprism, octagonal-truncated cubic duoprismatic prism, and the octagonal trioprism.

Vertex coordinates[edit | edit source]

The coordinates of a runcinated 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 runcinated 6-cube is the colonel of a regiment that includes 138 uniform members plus 15 fissaries.

External links[edit | edit source]