Medial hexacosichoron

Medial hexacosichoron
Rank	4
Type	Regular
Notation
Bowers style acronym	Mix
Elements
Components	120 pentachora
Cells	600 tetrahedra
Faces	1200 triangles
Edges	1200
Vertices	600
Vertex figure	Tetrahedron, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {10}}{5}}\approx 0.63246$
Inradius	${\frac {\sqrt {10}}{20}}\approx 0.15811$
Hypervolume	${\frac {5{\sqrt {5}}}{4}}\approx 2.79510$
Dichoral angle	$\arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }$
Related polytopes
Army	Hi
Regiment	Mix
Dual	Medial hexacosichoron
Conjugate	Medial hexacosichoron
Convex core	Hexacosichoron
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	H4, order 14400
Convex	No
Nature	Tame

The medial hexacosichoron, or mix, is a regular compound polychoron. It is a compound of 120 pentachora. It has 600 tetrahedra as cells, with 4 cells joining at each vertex. It can also be seen as a compound of 60 stellated decachora.

Vertex coordinates

The vertices of a medial hexacosichoron of edge length 1, centered at the origin, are given by all permutations of:

• $\left(\pm {\frac {\sqrt {5}}{5}},\,\pm {\frac {\sqrt {5}}{5}},\,0,\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5}}{10}},\,\pm {\frac {\sqrt {5}}{10}},\,\pm {\frac {\sqrt {5}}{10}}\right),$
• $\left(\pm {\frac {5+{\sqrt {5}}}{20}},\,\pm {\frac {5+{\sqrt {5}}}{20}},\,\pm {\frac {5+{\sqrt {5}}}{20}},\,\pm {\frac {3{\sqrt {5}}-5}{20}}\right),$
• $\left(\pm {\frac {5+3{\sqrt {5}}}{20}},\,\pm {\frac {5-{\sqrt {5}}}{20}},\,\pm {\frac {5-{\sqrt {5}}}{20}},\,\pm {\frac {5-{\sqrt {5}}}{20}}\right),$

together with all the even permutations of:

• $\left(\pm {\frac {\sqrt {5}}{10}},\,\pm {\frac {5+3{\sqrt {5}}}{20}},\,\pm {\frac {3{\sqrt {5}}-5}{20}},\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{20}},\,0,\,\pm {\frac {5-{\sqrt {5}}}{20}}\right),$
• $\left(\pm {\frac {\sqrt {5}}{10}},\,\pm {\frac {5+{\sqrt {5}}}{20}},\,\pm {\frac {\sqrt {5}}{5}},\,\pm {\frac {5-{\sqrt {5}}}{20}}\right).$

External links

• Klitzing, Richard. "mix".