Rhombicuboctahedral prism

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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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.png
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 Rhombicuboctahedron prism verf.png
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.


Small rhombicuboctahedral prism net.png
Rhombicuboctahedral prism.png
Schlegel diagram
One rhombicuboctahedron and triangular prisms show

Alternative names[edit]

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

Related polytopes[edit]

Runcic snub cubic hosochoron[edit]

Runcic snub cubic hosochoron
Schläfli symbol s3{2,4,3}
Coxeter diagram CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells 16 total:
2 t{3,3} Truncated tetrahedron.png
6 {3,3} Tetrahedron.png
8 tricup Triangular cupola.png
Faces 52 total:
32 {3}
8 {6}
Edges 60
Vertices 24
Vertex figure Runcic snub 243 verf.png
Symmetry group [4,3,2+], order 48
Properties convex

A related polychoron is the runcic snub cubic hosochoron, or parabidiminished rectified tesseract or truncated tetrahedral cupoliprism, s3{2,4,3}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, 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}, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 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 cupola connect the triangle and hexagonal faces, and the tetrahedral connect edge-wise between.

Runcic snub cubic hosochoron.png
(triangular cupolae hidden)
Truncated tetrahedral cupoliprism net.png


  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

External links[edit]