# Medial hexacosichoron

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Medial hexacosichoron
Rank4
TypeRegular
Notation
Bowers style acronymMix
Elements
Components120 pentachora
Cells600 tetrahedra
Faces1200 triangles
Edges1200
Vertices600
Vertex figureTetrahedron, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {10}}{5}}\approx 0.63246}$
Inradius${\displaystyle {\frac {\sqrt {10}}{20}}\approx 0.15811}$
Hypervolume${\displaystyle {\frac {5{\sqrt {5}}}{4}}\approx 2.79510}$
Dichoral angle${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
Related polytopes
ArmyHi
RegimentMix
DualMedial hexacosichoron
ConjugateMedial hexacosichoron
Convex coreHexacosichoron
Abstract & topological properties
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

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:

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

together with all the even permutations of:

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