Prismatorhombated hecatonicosachoron

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Prismatorhombated hecatonicosachoron
Runcitruncated 600-cell.png
Bowers style acronymPrahi
Coxeter diagramx5o3x3x (CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png)
Cells720 pentagonal prisms, 1200 hexagonal prisms, 600 truncated tetrahedra, 120 small rhombicosidodecahedra
Faces2400 triangles, 3600+3600 squares, 1440 pentagons, 2400 hexagons
Vertex figureIsosceles trapezoidal pyramid, base edge lengths 1, 2, (1+5)/2, 2; lateral edge lengths 2, 2, 3, 3
Measures (edge length 1)
Dichoral anglesTut–6–hip:
 Srid–5–pip: 162°
Central density1
Number of external pieces2640
Level of complexity16
Related polytopes
DualDeltopyramidal heptachiliadiacosichoron
ConjugatePrismatoquasirhombated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count230400
Euler characteristic0
SymmetryH4, order 14400

The prismatorhombated hecatonicosachoron, or prahi, also commonly called the runcitruncated 600-cell, is a convex uniform polychoron that consists of 720 pentagonal prisms, 1200 hexagonal prisms, 600 truncated tetrahedra, and 120 small rhombicosidodecahedra. 1 pentagonal prism, 2 hexagonal prisms, 1 truncated tetrahedron, and 1 small rhombicosidodecahedron join at each vertex. As one of its names suggests, it can be obtained by runcitruncating the hexacosichoron.

Cross-sections[edit | edit source]


Vertex coordinates[edit | edit source]

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

Semi-uniform variant[edit | edit source]

The prismatorhombated hecatonicosachoron has a semi-uniform variant of the form x5o3y3z that maintains its full symmetry. This variant uses 120 small rhombicosidodecahedra of form x5o3y, 600 truncated tetrahedra of form z3y3o, 720 pentagonal prisms of form z x5o, and 1200 ditrigonal prisms of form x y3z as cells, with 3 edge lengths.

With edges of length a, b, and c (such that it forms a5o3b3c), its circumradius is given by .

Related polychora[edit | edit source]

The prismatorhombated hecatonicosachoron is the colonel of a 3-member regiment that also includes the small prismatohexacosidishecatonicosachoron and the small rhombiprismic dishecatonicosachoron.

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]