Nonconvex great rhombicosidodecahedron

Template:Short description Template:Uniform polyhedra db File:Nonconvex great rhombicosidodecahedron.stl 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. It is also called the quasirhombicosidodecahedron. It is given a Schläfli symbol Template:Math 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 $$\begin{array}{ccccc}(& \pm \frac{1}{{\phi }^2},& 0,& \pm [2-\frac{1}{\phi }]& )\\ (& \pm 1,& \pm \frac{1}{{\phi }^3},& \pm 1& )\\ (& \pm \frac{1}{\phi },& \pm \frac{1}{{\phi }^2},& \pm \frac{2}{\phi }& )\end{array}$$

where $\phi =\frac{1+\sqrt{5}}{2}$ 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).

Nonconvex great rhombicosidodecahedron	Great dodecicosidodecahedron	Great rhombidodecahedron
Truncated great dodecahedron	Compound of six pentagonal prisms	Compound of twelve pentagonal prisms

Great deltoidal hexecontahedron

Template:Uniform polyhedra db File:Great deltoidal hexecontahedron.stl 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.

See also

• List of uniform polyhedra

References

• Template:Citation

External links

• Template:Mathworld
• Template:Mathworld
• Uniform polyhedra and duals