# Rectified hexacosichoron

(Redirected from Rox)
Rectified hexacosichoron Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymRox
Coxeter diagramo5o3x3o (       )
Elements
Cells600 octahedra, 120 icosahedra
Faces1200+2400 triangles
Edges3600
Vertices720
Vertex figurePentagonal prism, edge length 1
Edge figureike.oct.oct
Measures (edge length 1)
Circumradius$\sqrt{5+2\sqrt5} ≈ 3.07768$ Hypervolume$25\frac{31+15\sqrt5}{4} ≈ 403.38137$ Dichoral anglesOct–3–oct: $\arccos\left(-\frac{1+3\sqrt5}{8}\right) ≈ 164.47751^\circ$ Ike–3–oct: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$ Central density1
Number of external pieces720
Level of complexity3
Related polytopes
ArmyRox
RegimentRox
DualJoined hecatonicosachoron
ConjugateRectified grand hexacosichoron
Abstract & topological properties
Flag count43200
Euler characteristic0
OrientableYes
Properties
SymmetryH4, order 14400
ConvexYes
NatureTame

The rectified hexacosichoron, or rox, also commonly called the rectified 600-cell, is a convex uniform polychoron that consists of 600 regular octahedra and 120 regular icosahedra. Two icosahedra and 5 octahedra join at each pentagonal prismatic vertex. As the name suggests, it can be obtained by rectifying the hexacosichoron.

Blending 10 rectified hexacosichora results in the small disnub dishexacosichoron, which is uniform.

It is also isogonal under A3●H3 symmetry, where it can be called the snub tetrahedral hecatonicosachoron. In this symmetry the icosahedra have the symmetry of snub tetrahedra and 480 of the octahedra have triangular antiprismatic symmetry only. In fact each individual component in the blend of 10 rectified hexacosichora has this symmetry only.

It is also the vertex figure of the uniform hexadecachoric-hexacosichoric tetracomb, which is formed from alternating an order-5 tesseractic tetracomb.

## Vertex coordinates

The vertices of a rectified hexacosichoron of edge length 1 are given by all permutations of:

• $\left(0,\,0,\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{2}\right),$ • $\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$ along with even permutations of:

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

A rectified hexacosichoron has the following Coxeter diagrams:

• o5o3x3o (full symmetry)
• DCBAVFfxoo ooxooxxoof5oxoofoxxoo3xoxFofofVx5&#zx (H3×A1 symmetry)
• AoooFxoxVofoFofxxf5oAooxFxooVofoFxfxf ooAoxoFxofVoxfFofx5oooAoxxFfooVfxoFfx&#zx (H2×H2 symmetry)
• ooxooxxo(of)oxxooxoo5oxoofoxx(oo)xxofooxo3xoxFofof(Vx)fofoFxox&#xt (H3 axial, icosahedron-first)

## Related polychora

The rectified hexacosichoron is the colonel of a regiment with 15 members. Of these, one other besideds the colonel itself is Wythoffian (the rectified faceted hexacosichoron), two are hemi-Wythoffian (the small prismatohecatonicosachoron and small pentagonal retroprismatoverted dishecatonicosachoron), and one is noble (the small retropental hecatonicosachoron).

The segmentochoron icosahedron atop icosidodecahedron can be obtained as a cap of the rectified hexacosichoron in icosahedron-first orientation; the second segment in this orientation is icosidodecahedron atop small rhombicosidodecahedron.

It is also possible to diminish the rectified hexacosichoron by removing pentagonal prismatic pyramids from the vertices. If 120 vertices forming an inscribed hexacosichoron are removed, the result is the scaliform swirlprismatodiminished rectified hexacosichoron.

Uniform [[polychoron compound]s composed of rectified hexacosichora include:

### Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora: