Great rhombic triacontahedron

{{Short description|Polyhedron with 30 faces}}

{{Uniform polyhedra db|Uniform dual polyhedron stat table|gID}}

File:Great rhombic triacontahedron.stl

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 $\backslash varphi^3\; \backslash approx\; 4.236$, where $\backslash varphi\backslash !$ 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.

What resembles an "excavated" rhombic triacontahedron (compare excavated dodecahedron and excavated icosahedron) can be seen within the middle of this compound. The rest of the polyhedron strikingly resembles a rhombic hexecontahedron.

The rhombs have two angles of $\backslash arccos(\backslash frac\{1\}\{5\}\backslash sqrt\{5\})\backslash approx\; 63.434\backslash ,948\backslash ,822\backslash ,92^\{\backslash circ\}$, and two of $\backslash arccos(-\backslash frac\{1\}\{5\}\backslash sqrt\{5\})\backslash approx\; 116.565\backslash ,051\backslash ,177\backslash ,08^\{\backslash circ\}$. Its dihedral angles equal $\backslash arccos(-\backslash frac\{1\}\{4\}+\backslash frac\{1\}\{4\}\backslash sqrt\{5\})=\; 72^\{\backslash circ\}$. Part of each rhomb lies inside the solid, hence is invisible in solid models. The ratio between the lengths of the long and short diagonal of the rhombs equals the golden ratio $\backslash varphi$.

style="vertical-align: top;" |rowspan="2"| {{multiple image | align=left | perrow=2 | total_width=400 | image1 = Skeleton pair 12-20, size s.png | image2 = Rhombic triacontahedron 1 (convex), size s, pyritohedral.png | image3 = Skeleton pair Gr12 and dual, size s.png | image4 = Rhombic triacontahedron 2 (medial), pyritohedral.png | image5 = Skeleton pair Gr20 and dual, size s.png | image6 = Rhombic triacontahedron 3 (great), pyritohedral.png | footer = Convex, medial and great rhombic triacontahedron on the right (shown with pyritohedral symmetry) and the corresponding dual compounds of regular solids on the left }} | File:Rhombs of convex, medial and great rhombic triacontahedron.svg

{{multiple image | align = left | total_width = 400 | image1 = Rhombic triacontahedron 3 (great), size s, 2-fold.png | image2 = Rhombic triacontahedron 3 (great), size s, 3-fold.png | image3 = Rhombic triacontahedron 3 (great), size s, 5-fold.png | footer = Orthographic projections from 2-, 3- and 5-fold axes }}