Rectified hecatonicosachoron

Rectified hecatonicosachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Rahi
Coxeter diagram	o5x3o3o ()
Elements
Cells	600 tetrahedra, 120 icosidodecahedra
Faces	2400 triangles, 720 pentagons
Edges	3600
Vertices	1200
Vertex figure	Semi-uniform triangular prism, edge lengths 1 (base) and (1+√5)/2 (side)
Edge figure	tet 3 id 5 id 3
Measures (edge length 1)
Circumradius	$\frac{3\sqrt3+\sqrt{15}}{2} ≈ 4.53457$
Hypervolume	$5\frac{730+331\sqrt5}{4} ≈ 1837.67313$
Dichoral angles	Id–3–tet: $\arccos\left(-\frac{\sqrt{7+3\sqrt5}}{4}\right) ≈ 157.76124^\circ$
	Id–5–id: 144°
Central density	1
Number of external pieces	720
Level of complexity	3
Related polytopes
Army	Rahi
Regiment	Rahi
Dual	Joined hexacosichoron
Conjugate	Rectified great grand stellated hecatonicosachoron
Abstract & topological properties
Flag count	43200
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	Yes
Nature	Tame

The rectified hecatonicosachoron, or rahi, also commonly called the rectified 120-cell, is a convex uniform polychoron that consists of 600 regular tetrahedra and 120 icosidodecahedra. Two tetrahedra and three icosidodecahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the hecatonicosachoron.

Cross-sections

Vertex coordinates

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

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

along with all even permutations of:

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

Representations

A rectified hecatonicosachoron has the following Coxeter diagrams:

• o5x3o3o (full symmetry)
• ofxoxooxFf(oV)fFxooxoxfo5xoxfofFxoo(xo)ooxFfofxox3oooxFfofxF(Vo)FxfofFxooo&#xt (H₃ axial, icosidodecahedron-first)

Related polychora

The rectified hecatonicosachoron is the colonel of a 3-member regiment that also includes the facetorectified hecatonicosachoron and small hecatonicosintercepted hecatonicosachoron.

Isogonal derivatives

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

• Icosidodecahedron (120): Hexacosichoron
• Tetrahedron (600): Hecatonicosachoron
• Pentagon (720): Rectified hexacosichoron
• Triangle (2400): Semi-uniform small disprismatohexacosihecatonicosachoron
• Edge (3600): Semi-uniform small rhombated hecatonicosachoron

External links

• Bowers, Jonathan. "Category 3: Triangular Rectates" (#48).

• Klitzing, Richard. "rahi".

• Quickfur. "Rectified hecatonicosachoron".

• Wikipedia Contributors. "Rectified 120-cell".