Great rhombic triacontahedron

Great rhombic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Gort
Coxeter diagram	o5/2m3o ()
Elements
Faces	30 rhombi
Edges	60
Vertices	12+20
Vertex figure	20 triangles, 12 pentagrams
Measures (edge length 1)
Inradius	${\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\approx 0.32492$
Volume	$4{\sqrt {5-2{\sqrt {5}}}}\approx 2.90617$
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
Flag orbits	2
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 ${\frac {\sqrt {10\left(5-{\sqrt {5}}\right)}}{8}}\approx 0.65716$. ​The rhombus faces will have length ${\frac {\sqrt {5}}{2}}\approx 1.11803$, and width ${\frac {5-{\sqrt {5}}}{4}}\approx 0.69098$. The rhombi have two interior angles of $\arccos \left({\frac {\sqrt {5}}{5}}\right)\approx 63.43495^{\circ }$, and one of $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$.

Vertex coordinates

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

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

External links

• Klitzing, Richard. "gid".
• Wikipedia contributors. "Great rhombic triacontahedron".
• McCooey, David. "Great Rhombic Triacontahedron"