Truncated cuboctahedral prism

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Truncated cuboctahedral prism
Truncated cuboctahedral prism.png
Schlegel diagram
Type Prismatic uniform polychoron
Uniform index 55
Schläfli symbol t0,1,2,3{4,3,2} or tr{4,3}×{}
Coxeter-Dynkin CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
Cells 28 total:
2 Great rhombicuboctahedron.png 4.6.8
12 Hexahedron.png 4.4.4
8 Hexagonal prism.png 4.4.6
6 Octagonal prism.png 4.4.8
Faces 124 total:
96 {4}
16 {6}
12 {8}
Edges 192
Vertices 96
Vertex figure Truncated cuboctahedral prism vertex figure.png
Irregular tetrahedron
Symmetry group [4,3,2], order 96
Properties convex

In geometry, a truncated cuboctahedral prism or great rhombicuboctahedral prism is a convex uniform polychoron (four-dimensional polytope).

It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.

Great rhombicuboctahedral prism net.png
Net

Alternative names[]

  • Truncated-cuboctahedral dyadic prism (Norman W. Johnson)
  • Gircope (Jonathan Bowers: for great rhombicuboctahedral prism/hyperprism)
  • Great rhombicuboctahedral prism/hyperprism

Related polytopes[]

A full snub cubic antiprism or omnisnub cubic antiprism can be defined as an alternation of an truncated cuboctahedral prism, represented by ht0,1,2,3{4,3,2}, or CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png, although it cannot be constructed as a uniform polychoron. It has 76 cells: 2 snub cubes connected by 12 tetrahedrons, 6 square antiprisms, and 8 octahedrons, with 48 tetrahedrons in the alternated gaps. There are 48 vertices, 192 edges, and 220 faces (12 squares, and 16+192 triangles). It has [4,3,2]+ symmetry, order 48.

A construction exists with two uniform snub cubes in snub positions with two edge lengths in a ratio of around 1 : 1.138.

Omnisnub cubic antiprism vertex figure.png
Vertex figure for the omnisnub cubic antiprism

Also related is the bialternatosnub octahedral hosochoron, constructed by removing alternating long rectangles from the octagons, but is also not uniform. It has 40 cells: 2 rhombicuboctahedra (with Th symmetry), 6 rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), 8 octahedra (as triangular antiprisms), 24 triangular prisms (as Cs-symmetry wedges) filling the gaps, and 48 vertices. It has [4,(3,2)+] symmetry, order 48. Its vertex figure is a chiral hexahedron topologically identical to the tetragonal antiwedge.

Bialternatosnub octahedral hosochoron vertex figure.png
Vertex figure for the bialternatosnub octahedral hosochoron

External links[]

  • 6. Convex uniform prismatic polychora - Model 55, George Olshevsky.
  • Klitzing, Richard. "4D uniform polytopes (polychora) x3x4x x - gircope".


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