Small rhombated hecatonicosachoron

Small rhombated hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Srahi
Coxeter diagram	x5o3x3o ()
Elements
Cells	600 octahedra, 1200 triangular prisms, 120 small rhombicosidodecahedra
Faces	2400+2400 triangles, 3600 squares, 720 pentagons
Edges	3600+7200
Vertices	3600
Vertex figure	Square wedge, edge lengths 1 (base square), √2 (side edges), and (1+√5)/2 (top edge)
Measures (edge length 1)
Circumradius	$\sqrt{23+10\sqrt5} \approx 6.73503$
Hypervolume	$5\frac{1870+823\sqrt5}{2} \approx 9275.70986$
Dichoral angles	Oct–3–trip: $\arccos\left(-\frac{\sqrt6+\sqrt{30}}{8}\right) \approx 172.23876^\circ$
	Srid–4–trip: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) \approx 159.09484^\circ$
	Srid–3–oct: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) \approx 157.76124^\circ$
	Srid–5–srid: 144°
Central density	1
Number of external pieces	1920
Level of complexity	9
Related polytopes
Army	Srahi
Regiment	Srahi
Dual	Great notched trischiliahexacosichoron
Conjugate	Quasirhombated great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count	129600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

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

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

• $\left(0,\,0,\,±(2+\sqrt5),\,±(3+\sqrt5)\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac12,\,±\frac12,\,±\frac{5+2\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±(2+\sqrt5),\,±(2+\sqrt5)\right),$
• $\left(±\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2},\,±\frac{3+2\sqrt5}{2},\,±\frac{5+2\sqrt5}{2}\right),$

together with all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{13+5\sqrt5}{4},\,±\frac{5+3\sqrt5}{4}\right),$
• $\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac{11+5\sqrt5}{4},\,±\frac{3+2\sqrt5}{2}\right),$
• $\left(0,\,±\frac{3+\sqrt5}{4},\,±3\frac{2+\sqrt5}{2},\,±\frac{5+\sqrt5}{4}\right),$
• $\left(0,\,±\frac{2+\sqrt5}{2},\,±\frac{9+5\sqrt5}{4},\,±3\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{9+5\sqrt5}{4},\,±(2+\sqrt5)\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{13+5\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$
• $\left(±\frac12,\,±\frac{1+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±\frac{7+3\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{7+3\sqrt5}{4},\,±(2+\sqrt5),\,±3\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±3\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{2},\,±\frac{5+2\sqrt5}{2},\,±3\frac{3+\sqrt5}{4}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{13+5\sqrt5}{4},\,±\frac{2+\sqrt5}{2}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±(2+\sqrt5)\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±(3+\sqrt5),\,±\frac{7+3\sqrt5}{4}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{7+3\sqrt5}{4},\,±\frac{3+2\sqrt5}{2},\,±(2+\sqrt5)\right),$
• $\left(±\frac{1+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{5+2\sqrt5}{2},\,±\frac{7+3\sqrt5}{4}\right),$
• $\left(±\frac{5+\sqrt5}{4},\,±\frac{2+\sqrt5}{2},\,±\frac{11+5\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$
• $\left(±\frac{2+\sqrt5}{2},\,±\frac{3+\sqrt5}{2},\,±\frac{5+3\sqrt5}{4},\,±\frac{9+5\sqrt5}{4}\right).$

Semi-uniform variant

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 $\sqrt{7a^2+5b^2+11ab+(3a^2+2b^2+5ab)\sqrt5}$.

Related polychora

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.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#158).

• Klitzing, Richard. "srahi".

• Quickfur. "The Cantellated 120-cell".

• Wikipedia Contributors. "Cantellated 120-cell".