# Medial icosacronic hexecontahedron

Medial icosacronic hexecontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramo5/3m3m5*a
Elements
Faces60 darts
Edges60+60
Vertices12+12+20
Vertex figure12 pentagons, 12 pentagrams, 20 hexagons
Measures (edge length 1)
Inradius$\frac{3\sqrt7}{7} ≈ 1.13389$ Dihedral angle$\arccos\left(-\frac57\right) ≈ 135.58469°$ Central density4
Number of external pieces180
Related polytopes
Abstract & topological properties
Flag count480
Euler characteristic–16
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The medial icosacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

If its dual, the icosidodecadodecahedron, has an edge length of 1, then the short edges of the darts will measure $\frac{7\sqrt6-\sqrt{30}}{11} ≈ 1.06084$ , and the long edges will be $\frac{7\sqrt6+\sqrt{30}}{11} ≈ 2.05670$ . ​The dart faces will have length $\frac{6\sqrt7}{11} ≈ 1.44314$ , and width 2. ​The darts have two interior angles of $\arccos\left(\frac34\right) ≈ 41.40962°$ , one of $\arccos\left(-\frac18+\frac{7\sqrt5}{24}\right) ≈ 58.18445°$ , and one of $360°-\arccos\left(-\frac18-\frac{7\sqrt5}{24}\right) ≈ 218.99631°$ .

## Vertex coordinates

A medial icosacronic hexecontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• $\left(±\frac{\sqrt5-1}{2},\,±\frac{1+\sqrt5}{2},\,0\right),$ • $\left(±3\frac{7+\sqrt5}{22},\,±3\frac{3\sqrt5-1}{22},\,0\right),$ • $\left(±3\frac{1+3\sqrt5}{22},\,±3\frac{7-\sqrt5}{22},\,0\right),$ • $\left(±1,\,±1,\,±1\right).$ 