Nonconvex great rhombicosidodecahedron explained

In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U₆₇. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices.^[1] It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol Its vertex figure is a crossed quadrilateral.

This model shares the name with the convex great rhombicosidodecahedron, also known as the truncated icosidodecahedron.

Cartesian coordinates

Cartesian coordinates for the vertices of a nonconvex great rhombicosidodecahedron are all the even permutations of $\begin \Bigl(& \pm\,\frac,& 0,& \pm \bigl[2-\frac{1}{\varphi}\bigr] &\Bigr) \\ \Bigl(& \pm\,1,& \pm\,\frac,& \pm\,1 &\Bigr) \\ \Bigl(& \pm\,\frac,& \pm\,\frac,& \pm\,\frac &\Bigr)\end$

where

\varphi=\tfrac{1+\sqrt5}{2}

is the golden ratio.

Related polyhedra

It shares its vertex arrangement with the truncated great dodecahedron, and with the uniform compounds of 6 or 12 pentagonal prisms. It additionally shares its edge arrangement with the great dodecicosidodecahedron (having the triangular and pentagrammic faces in common), and the great rhombidodecahedron (having the square faces in common).

Great deltoidal hexecontahedron

The great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices.

It is also called a great strombic hexecontahedron.

External links

- - Uniform polyhedra and duals

Notes and References

Web site: 67: great rhombicosidodecahedron. Maeder. Roman. MathConsult.