Prismatorhombated hexacosichoron

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Prismatorhombated hexacosichoron
Runcitruncated 120-cell.png
Bowers style acronymPrix
Coxeter diagramx5x3o3x (CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Cells1200 triangular prisms, 720 decagonal prisms, 600 cuboctahedra, 120 truncated dodecahedra
Faces2400+2400 triangles, 3600+3600 squares, 1440 decagons
Vertex figureSkewed rectangular pyramid, base edge lengths 1, 2, 1, 2; lateral edge lengths 2, 2, (5+5)/2, (5+5)/2
Measures (edge length 1)
Dichoral anglesCo–3–trip:
 Tid–10–dip: 162°
Central density1
Number of external pieces2640
Level of complexity16
Related polytopes
DualRhombipyramidal heptachiliadiacosichoron
ConjugateQuasiprismatorhombated grand hexacosichoron
Abstract & topological properties
Flag count230400
Euler characteristic0
SymmetryH4, order 14400

The prismatorhombated hexacosichoron, or prix, also commonly called the runcitruncated 120-cell, is a convex uniform polychoron that consists of 1200 triangular prisms, 720 decagonal prisms, 600 cuboctahedra, and 120 truncated dodecahedra. 1 triangular prism, 2 decagonal prisms, 1 cuboctahedron, and 1 truncated dodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hecatonicosachoron.

Cross-sections[edit | edit source]


Vertex coordinates[edit | edit source]

The vertices of a prismatorhombated hexacosichoron of edge length 1 are given by all permutations of:

Plus all even permutations of:

Semi-uniform variant[edit | edit source]

The prismatorhombated hexacosichoron has a semi-uniform variant of the form x5y3o3z that maintains its full symmetry. This variant uses 120 truncated dodecahedra of form x5y3o, 600 rhombitetratetrahedra of form y3o3z, 1200 triangular prisms of form x z3o, and 720 dipentagonal prisms of form z x5y as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a5b3o3c), its circumradius is given by .

Related polychora[edit | edit source]

The prismatorhombated hexacosichoron is the colonel of a 3-member regiment that also includes the small prismatohexacosihecatonicosihecatonicosachoron and the small rhombiprismic hexacosihecatonicosachoron.

The segmentochoron truncated dodecahedron atop great rhombicosidodecahedron can be obtained as a cap of the prismatorhombated hexacosichoron.

o5o3o3o truncations
Name OBSA CD diagram Picture
Hecatonicosachoron hi x5o3o3o
Schlegel wireframe 120-cell.png
Truncated hecatonicosachoron thi x5x3o3o
Schlegel half-solid truncated 120-cell.png
Rectified hecatonicosachoron rahi o5x3o3o
Hexacosihecatonicosachoron xhi o5x3x3o
Rectified hexacosichoron rox o5o3x3o
Rectified 600-cell schlegel halfsolid.png
Truncated hexacosichoron tex o5o3x3x
Schlegel half-solid truncated 600-cell.png
Hexacosichoron ex o5o3o3x
Schlegel wireframe 600-cell.png
Small rhombated hecatonicosachoron srahi x5o3x3o
Great rhombated hecatonicosachoron grahi x5x3x3o
Cantitruncated 120-cell.png
Small rhombated hexacosichoron srix o5x3o3x
Great rhombated hexacosichoron grix o5x3x3x
Cantitruncated 600-cell.png
Small disprismatohexacosihecatonicosachoron sidpixhi x5o3o3x
Runcinated 120-cell.png
Prismatorhombated hexacosichoron prix x5x3o3x
Runcitruncated 120-cell.png
Prismatorhombated hecatonicosachoron prahi x5o3x3x
Runcitruncated 600-cell.png
Great disprismatohexacosihecatonicosachoron gidpixhi x5x3x3x
Omnitruncated 120-cell wireframe.png

External links[edit | edit source]