Small rhombated hecatonicosachoron

From Polytope Wiki
Jump to navigation Jump to search
Small rhombated hecatonicosachoron
Bowers style acronymSrahi
Coxeter diagramx5o3x3o (CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png)
Cells600 octahedra, 1200 triangular prisms, 120 small rhombicosidodecahedra
Faces2400+2400 triangles, 3600 squares, 720 pentagons
Vertex figureSquare wedge, edge lengths 1 (base square), 2 (side edges), and (1+5)/2 (top edge)
Measures (edge length 1)
Dichoral anglesOct–3–trip:
 Srid–5–srid: 144°
Central density1
Number of external pieces1920
Level of complexity9
Related polytopes
DualGreat notched trischiliahexacosichoron
ConjugateQuasirhombated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count129600
Euler characteristic0
SymmetryH4, order 14400

The small rhombated hecatonicosachoron, or srahi, also commonly called the cantellated 120-cell, is a convex uniform polychoron that consists of 600 regular octahedra, 1200 triangular prisms, and 120 small rhombicosidodecahedra. 1 octahedron, 2 triangular prisms, and 2 small rhombicosidodecahedra join at each vertex. As one of its names suggests, it can be obtained by cantellating the hecatonicosachoron.

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a small rhombated hecatonicosachoron of edge length 1 are given by all permutations of:

together with all even permutations of:

Semi-uniform variant[edit | edit source]

The small rhombated hecatonicosachoron has a semi-uniform variant of the form x5o3y3o that maintains its full symmetry. This variant uses 600 octahedra of size y, 120 semi-uniform small rhombicosidodecahedra of form x5o3y, and 1200 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by two small rhombicosidodecahedra) and b (of octahedra), its circumradius is given by .

Related polychora[edit | edit source]

The small rhombated hecatonicosachoron is the colonel of a seven-member regiment. Its other members include the small retrosphenoverted hexacosidishecatonicosachoron, small rhombic dishecatonicosachoron, small pseudorhombic prismatohecatonicosachoron, grand rhombic prismatohecatonicosachoron, prismatohecatonicosintercepted hexacosihecatonicosachoron, and prismatointercepted prismatohexacosihecatonicosachoron.

The segmentochoron small rhombicosidodecahedron atop truncated dodecahedron can be obtained as a cap of the small rhombated hecatonicosachoron. Another possible cap is the pentagonal pucofastegium, viewed as a cap from a pentagonal face.

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]