Small rhombated hexacosichoron

Small rhombated hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Srix
Coxeter diagram	o5x3o3x ()
Elements
Cells	600 cuboctahedra, 720 pentagonal prisms, 120 icosidodecahedra
Faces	1200+2400 triangles, 3600 squares, 1440 pentagons
Edges	3600+7200
Vertices	3600
Vertex figure	Rectangular 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)
Circumradius	$\sqrt{19+8\sqrt5} ≈ 6.07359$
Hypervolume	$5\frac{1240+591\sqrt5}{2} ≈ 6403.79044$
Dichoral angles	Co–4–pip: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{10}}\right) ≈ 166.71747^\circ$
	Co–3–co: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751^\circ$
	Id–5–pip: 162°
	Id–3–co: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
Central density	1
Number of external pieces	1440
Level of complexity	9
Related polytopes
Army	Srix
Regiment	Srix
Dual	Small notched trischiliahexacosichoron
Conjugate	Small rhombated grand hexacosichoron
Abstract & topological properties
Flag count	129600
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

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

Vertex coordinates

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

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

together with all even permutations of:

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

Semi-uniform variant

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 $\sqrt{\frac{3a^2+21b^2+14ab+(a^2+9b^2+6ab)\sqrt5}{2}}$.

Related polychora

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.

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#165).

• Klitzing, Richard. "srix".

• Quickfur. "The Cantellated 600-cell".

• Wikipedia Contributors. "Cantellated 600-cell".