# Great rhombic triacontahedron

Great rhombic triacontahedron Rank3
TypeUniform dual
SpaceSpherical
Notation
Bowers style acronymGort
Coxeter diagramo5/2m3o
Elements
Faces30 rhombi
Edges60
Vertices20+12
Vertex figure20 triangles, 12 pentagrams
Measures (edge length 1)
Inradius$\sqrt{\frac{5-2\sqrt5}{5}} ≈ 0.32492$ Volume$4\sqrt{5-2\sqrt5} ≈ 2.90617$ Dihedral angle72°
Central density7
Number of external pieces180
Related polytopes
DualGreat icosidodecahedron
ConjugateRhombic triacontahedron
Convex hullDodecahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count240
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The great rhombic triacontahedron is a uniform dual polyhedron. It consists of 30 rhombi.

If its dual, the great icosidodecahedron, has an edge length of 1, then the edges of the rhombi will measure $\frac{\sqrt{10\left(5-\sqrt5\right)}}{8} ≈ 0.65716$ . ​The rhombus faces will have length $\frac{\sqrt5}{2} ≈ 1.11803$ , and width $\frac{5-\sqrt5}{4} ≈ 0.69098$ . The rhombi have two interior angles of $\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495°$ , and one of $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$ .

## Vertex coordinates

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

• $\left(±\frac{5-\sqrt5}{8},\,±\frac{3\sqrt5-5}{8},\,0\right),$ • $\left(±\frac{3\sqrt5-5}{8},\,±\frac{\sqrt5}{4},\,0\right),$ • $\left(±\frac{5-\sqrt5}{8},\,±\frac{5-\sqrt5}{8},\,±\frac{5-\sqrt5}{8}\right).$ 