# Small icosacronic hexecontahedron

Small icosacronic hexecontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5/2o3m3*a
Elements
Faces60 kites
Edges60+60
Vertices20+12+20
Vertex figure20 triangles, 12 pentagrams, 20 hexagons
Measures (edge length 1)
Inradius$3\frac{\sqrt{305\left(9+2\sqrt5\right)}}{122} ≈ 1.57627$ Dihedral angle$\arccos\left(-\frac{44+3\sqrt5}{61}\right) ≈ 146.23066°$ Central density2
Number of external pieces120
Related polytopes
DualSmall icosicosidodecahedron
Abstract & topological properties
Flag count480
Euler characteristic–8
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small icosacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 kites.

If its dual, the small icosicosidodecahedron, has an edge length of 1, then the short edges of the kites will measure $\frac{\sqrt{30\left(65-19\sqrt5\right)}}{22} ≈ 1.18133$ , and the long edges will be $\frac{\sqrt{30\left(85-\sqrt5\right)}}{38} ≈ 1.31129$ . ​The kite faces will have length $3\frac{\sqrt{10\left(3517-585\sqrt5\right)}}{418} ≈ 1.06668$ , and width $\sqrt5 ≈ 2.23607$ . ​The kites have two interior angles of $\arccos\left(\frac34-\frac{\sqrt5}{20}\right) ≈ 50.34252°$ , one of $\arccos\left(-\frac{1}{12}-\frac{19\sqrt5}{60}\right) ≈ 142.31856°$ , and one of $\arccos\left(-\frac{5}{12}-\frac{\sqrt5}{60}\right) ≈ 116.99640°$ .

## Vertex coordinates

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

• $\left(±\frac{5-\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,0\right),$ • $\left(±3\frac{5+7\sqrt5}{44},\,±3\frac{15-\sqrt5}{44},\,0\right),$ • $\left(±\frac{\sqrt5}{2},\,±\frac{\sqrt5}{2},\,±\frac{\sqrt5}{2}\right),$ • $\left(±3\frac{9\sqrt5-5}{76},\,±3\frac{15+11\sqrt5}{76},\,0\right),$ • $\left(±3\frac{10+\sqrt5}{38},\,±3\frac{10+\sqrt5}{38},\,±3\frac{10+\sqrt5}{38}\right).$ 