Small rhombated hexacosichoron

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Small rhombated hexacosichoron
Bowers style acronymSrix
Coxeter diagramo5x3o3x (CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png)
Cells600 cuboctahedra, 720 pentagonal prisms, 120 icosidodecahedra
Faces1200+2400 triangles, 3600 squares, 1440 pentagons
Vertex figureRectangular wedge, edge lengths 1 (two edges of base rectangle and top edge), 2 (side edges), and (1+5)/2 (remaining base edges)
Measures (edge length 1)
Dichoral anglesCo–4–pip:
 Id–5–pip: 162°
Central density1
Number of external pieces1440
Level of complexity9
Related polytopes
DualSmall notched trischiliahexacosichoron
ConjugateSmall rhombated grand hexacosichoron
Abstract & topological properties
Flag count129600
Euler characteristic0
SymmetryH4, order 14400

The small rhombated hexacosichoron, or srix, also commonly called the cantellated 600-cell, is a convex uniform polychoron that consists of 600 cuboctahedra, 720 pentagonal prisms, and 120 icosidodecahedra. 1 icosidodecahedron, 2 pentagonal prisms, and 2 cuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantellating the hexacosichoron.

Gallery[edit | edit source]


Vertex coordinates[edit | edit source]

Coordinates for the vertices of a small rhombated hexacosichoron 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 hexacosichoron has a semi-uniform variant of the form o5y3o3x that maintains its full symmetry. This variant uses 120 icosidodecahedra of size y, 600 rhombitetratetrahedra of form x3o3y, and 720 pentagonal prisms of form x y5o as cells, with 2 edge lengths.

With edges of length a (surrounds 2 rhombitetratetrahedra) and b (of icosidodecahedra), its circumradius is given by .

Related polychora[edit | edit source]

The small rhombated hexacosichoron is the colonel of a seven-member regiment. Its other members include the small retrosphenoverted hecatonicosihexacosihecatonicosachoron, rhombic small hexacosihecatonicosachoron, pseudorhombic small dishecatonicosachoron, grand rhombic small hexacosihecatonicosachoron, small dishecatonicosintercepted hexacosihecatonicosachoron, and hecatonicosintercepted prismatohexacosihecatonicosachoron.

The segmentochoron icosidodecahedron atop truncated icosahedron can be obtained as a cap of the small rhombated 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]