Great deltoidal hexecontahedron

The great deltoidal hexecontahedron is a uniform dual polyhedron. It consists of 60 darts.

Great deltoidal hexecontahedron
Rank3
TypeUniform dual
SpaceSpherical
Notation
Coxeter diagramm5/3o3m
Elements
Faces60 darts
Edges60+60
Vertices20+30+12
Vertex figure20 triangles, 30 squares, 12 pentagrams
Measures (edge length 1)
Inradius${\displaystyle \frac{\sqrt{205\left(19−8\sqrt5\right)}}{41} ≈ 0.36816}$
Dihedral angle${\displaystyle \arccos\left(-\frac{19−8\sqrt5}{41}\right) ≈ 91.55340°}$
Central density13
Number of external pieces120
Related polytopes
DualQuasirhombicosidodecahedron
Abstract & topological properties
Flag count480
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

It appears the same as the great rhombidodecacron.

If its dual, the quasirhombicosidodecahedron, has an edge length of 1, then the short edges of the darts will measure ${\displaystyle \frac{\sqrt{5\left(5+\sqrt5\right)}}{3} ≈ 2.00500}$, and the long edges will be ${\displaystyle \frac{\sqrt{5\left(85+31\sqrt5\right)}}{11} ≈ 2.52523}$. ​The dart faces will have length ${\displaystyle \frac{\sqrt{10\left(157-31\sqrt5\right)}}{33} ≈ 0.89731}$, and width ${\displaystyle \frac{5+\sqrt5}{2} ≈ 3.61803}$. The darts have two interior angles of ${\displaystyle \arccos\left(\frac12+\frac{\sqrt5}{5}\right) ≈ 18.69941°}$, one of ${\displaystyle \arccos\left(-\frac14+\frac{\sqrt5}{10}\right) ≈ 91.51239°}$, and one of ${\displaystyle 360°-\arccos\left(-\frac18-\frac{9\sqrt5}{40}\right) ≈ 231.08879°}$.

Vertex coordinates

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

• ${\displaystyle \left(±\sqrt5,\,0,\,0\right),}$
• ${\displaystyle \left(±\frac{25-9\sqrt5}{22},\,±\frac{15-\sqrt5}{22},\,0\right),}$
• ${\displaystyle \left(±\frac{5-\sqrt5}{6},\,±\frac{3\sqrt5-5}{6},\,0\right),}$
• ${\displaystyle \left(±\frac{\sqrt5}{2},\,±\frac{5+\sqrt5}{4},\,±\frac{5-\sqrt5}{4}\right),}$
• ${\displaystyle \left(±\frac{4\sqrt5-5}{11},\,±\frac{4\sqrt5-5}{11},\,±\frac{4\sqrt5-5}{11}\right).}$