Great rhombic triacontahedron

Great rhombic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Space	Spherical
Notation
Bowers style acronym	Gort
Coxeter diagram	o5/2m3o
Elements
Faces	30 rhombi
Edges	60
Vertices	20+12
Vertex figure	20 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 angle	72°
Central density	7
Number of external pieces	180
Related polytopes
Dual	Great icosidodecahedron
Conjugate	Rhombic triacontahedron
Convex hull	Dodecahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	240
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

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:

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

External links

• Klitzing, Richard. "gid".

• Wikipedia Contributors. "Great rhombic triacontahedron".
• McCooey, David. "Great Rhombic Triacontahedron"