Faceted hexacosichoron

Faceted hexacosichoron
(OFF file)
Rank	4
Type	Regular
Notation
Bowers style acronym	Fix
Coxeter diagram	o5/2o5o3x ()
Schläfli symbol	$\{3,5,5\mid 3\}$ $\{3,5,5/2\}$
Elements
Cells	120 icosahedra
Faces	1200 triangles
Edges	720
Vertices	120
Vertex figure	Great dodecahedron, edge length 1
Edge figure	ike 3 ike 3 ike 3 ike 3 ike 3
Deep holes	Triangles
Measures (edge length 1)
Circumradius	${\frac {1+{\sqrt {5}}}{2}}\approx 1.61803$
Edge radius	${\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 1.53884$
Face radius	${\frac {3{\sqrt {3}}+{\sqrt {15}}}{6}}\approx 1.51152$
Inradius	${\frac {3+{\sqrt {5}}}{4}}\approx 1.30902$
Hypervolume	$25{\frac {7+3{\sqrt {5}}}{4}}\approx 85.67627$
Dichoral angle	120°
Central density	4
Number of external pieces	2400
Level of complexity	4
Related polytopes
Army	Ex
Regiment	Ex
Company	Ex
Dual	Small stellated hecatonicosachoron
Conjugate	Great faceted hexacosichoron
Convex core	Hecatonicosachoron
Abstract & topological properties
Flag count	14400
Euler characteristic	480
Schläfli type	{3,5,5}
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The faceted hexacosichoron, or fix, also commonly called the icosahedral 120-cell, is one of the 10 Schläfli–Hess polychora. It has 120 icosahedra as cells, joining 5 to an edge and 12 to a vertex in the form of a great dodecahedron.

As the name suggests, it is a faceting of the hexacosichoron, sharing its vertices, edges, and faces. The icosahedral cells are the same as the vertex figures of the hexacosichoron.

Cross-sections

Vertex coordinates

Its vertices are the same as those of its regiment colonel, the hexacosichoron.

Related polychora

Uniform polychoron compounds composed of faceted hexacosichora include:

• Faceted chirotetrahedral dishexacosichoron (2)
• Faceted snub pentishecatonicosachoron (5)
• Chiricosahedral hyperprismatochoron (6)
• Faceted snub decahecatonicosachoron (10)
• Disicosahedral hyperprismatochoron (12)

External links

• Bowers, Jonathan. "Category 1: Regular Polychora" (#7).

• Klitzing, Richard. "fix".
• Nan Ma. "Icosahedral 120-cell {3, 5, 5/2}".

• Wikipedia contributors. "Icosahedral 120-cell".