# Great rhombic triacontahedron

Great rhombic triacontahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymGort
Coxeter diagramo5/2m3o ()
Elements
Faces30 rhombi
Edges60
Vertices12+20
Vertex figure20 triangles, 12 pentagrams
Measures (edge length 1)
Inradius${\displaystyle {\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\approx 0.32492}$
Volume${\displaystyle 4{\sqrt {5-2{\sqrt {5}}}}\approx 2.90617}$
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
Flag orbits2
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 ${\displaystyle {\frac {\sqrt {10\left(5-{\sqrt {5}}\right)}}{8}}\approx 0.65716}$. ​The rhombus faces will have length ${\displaystyle {\frac {\sqrt {5}}{2}}\approx 1.11803}$, and width ${\displaystyle {\frac {5-{\sqrt {5}}}{4}}\approx 0.69098}$. The rhombi have two interior angles of ${\displaystyle \arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }}$, and one of ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{8}},\,\pm {\frac {3{\sqrt {5}}-5}{8}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {3{\sqrt {5}}-5}{8}},\,\pm {\frac {\sqrt {5}}{4}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{8}},\,\pm {\frac {5-{\sqrt {5}}}{8}},\,\pm {\frac {5-{\sqrt {5}}}{8}}\right)}$.