Rhombicuboctahedral prism

Rhombicuboctahedral prism
Type Prismatic uniform polychoron
Uniform index 53
Schläfli symbol t0,2,3{3,4,2} or rr{3,4}×{}
s2,3{3,4,2} or s2{3,4}×{}
Coxeter diagram
Cells 28 total:
2 rr{4,3} or s2{3,4}
8 {}x{3}
18 {4,3}
Faces 100 total:
16 {3}
84 {4}
Edges 120
Vertices 48
Vertex figure
Trapezoidal pyramid
Symmetry group [4,3,2], order 96
[3+,4,2], order 48
Properties convex

In geometry, a 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.

Images

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Net
 
Schlegel diagram
One rhombicuboctahedron and triangular prisms show

Alternative names

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  • small rhombicuboctahedral prism
  • (Small) rhombicuboctahedral dyadic prism (Norman W. Johnson)
  • Sircope (Jonathan Bowers: for small-rhombicuboctahedral prism)
  • (small) rhombicuboctahedral hyperprism
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Runcic snub cubic hosochoron

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Runcic snub cubic hosochoron
Schläfli symbol s3{2,4,3}
Coxeter diagram        
Cells 16 total:
2 t{3,3}  
6 {3,3}  
8 tricup  
Faces 52 total:
32 {3}
12{4}
8 {6}
Edges 60
Vertices 24
Vertex figure  
Symmetry group [4,3,2+], order 48
Properties convex

A related polychoron is the runcic snub cubic hosochoron, also known as a parabidiminished rectified tesseract, truncated tetrahedral alterprism, or truncated tetrahedral cupoliprism, s3{2,4,3},        . It is made from 2 truncated tetrahedra, 6 tetrahedra, and 8 triangular cupolae in the gaps, for a total of 16 cells, 52 faces, 60 edges, and 24 vertices. It is vertex-transitive, and equilateral, but not uniform, due to the cupolae. It has symmetry [2+,4,3], order 48.[1][2][3]

It is related to the 16-cell in its s{2,4,3},         construction.

It can also be seen as a prismatic polytope with two parallel truncated tetrahedra in dual positions, as seen in the compound of two truncated tetrahedra. Triangular cupolae connect the triangular and hexagonal faces, and the tetrahedral connect edge-wise between.

 
Projection
(triangular cupolae hidden)
 
Net

References

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  1. ^ Klitzing, Richard. "4D tutcup".
  2. ^ Category S1: Simple Scaliforms Tutcup
  3. ^ http://bendwavy.org/klitzing/pdf/artConvSeg_8.pdf 4.55 truncated tetrahedron || inverse truncated tetrahedron
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