Great rhombic triacontahedron

Great rhombic triacontahedron
Type Star polyhedron
Face
Elements F = 30, E = 60
V = 32 (χ = 2)
Symmetry group Ih, [5,3], *532
Index references DU54
dual polyhedron Great icosidodecahedron

In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also 20 on 3-fold and 12 on 5-fold axes).

It can be constructed from the convex solid by expanding the faces by factor of ${\displaystyle \varphi ^{3}\approx 4.236}$, where ${\displaystyle \varphi \!}$ is the golden ratio.

This solid is to the compound of great icosahedron and great stellated dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs.

 @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}}Convex, medial and great rhombic triacontahedron with the same face centers (shown with pyritohedral symmetry) The diagonal lengths of the three rhombic triacontahedra are powers of ${\displaystyle \varphi }$. With the Kepler-Poinsot compound Orthographic projections from 2-, 3- and 5-fold symmetry axes

References

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208