Rectified hexacosichoron

Rectified hexacosichoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Rox
Coxeter diagram	o5o3x3o ()
Elements
Cells	600 octahedra, 120 icosahedra
Faces	1200+2400 triangles
Edges	3600
Vertices	720
Vertex figure	Pentagonal prism, edge length 1
Edge figure	ike.oct.oct
Measures (edge length 1)
Circumradius	$\sqrt{5+2\sqrt5} ≈ 3.07768$
Hypervolume	$25\frac{31+15\sqrt5}{4} ≈ 403.38137$
Dichoral angles	Oct–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 density	1
Number of external pieces	720
Level of complexity	3
Related polytopes
Army	Rox
Regiment	Rox
Dual	Joined hecatonicosachoron
Conjugate	Rectified grand hexacosichoron
Abstract & topological properties
Flag count	43200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

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.

Cross-sections

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:

• Rectified chirotetrahedral dishexacosichoron (2)
• Rectified snub pentishecatonicosachoron (5)

Isogonal derivatives

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

• Icosahedron (120): Hexacosichoron
• Octahedron (600): Hecatonicosachoron
• Triangle (1200): Rectified hecatonicosachoron
• Triangle (2400): Semi-uniform small disprismatohexacosihecatonicosachoron
• Edge (3600): Uniform small rhombated hexacosichoron

External links

• Bowers, Jonathan. "Category 5: Pentagonal Rectates" (#73).

• Klitzing, Richard. "rox".

• Quickfur. "The Rectified 600-cell".

• Wikipedia Contributors. "Rectified 600-cell".