In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of snub polyhedra.

Cartesian coordinates Let be the real zero of the polynomial . The constant is known as the plastic ratio. Denote by the golden ratio.

Let the point be given by

Let the matrix be given by

is the rotation around the axis by an angle of , counterclockwise. Let the linear transformations be the transformations which send a point to the even permutations of with an even number of minus signs. The transformations constitute the group of rotational symmetries of a regular tetrahedron. The transformations , constitute the group of rotational symmetries of a regular icosahedron. Then the 60 points are the vertices of a snub icosidodecadodecahedron.

The edge length , the circumradius equals and the midradius To get a snub icosidodecadodecahedron with unit edge length, divide all above coordinates by .

Properties The snub icosidodecadodecahedron was among the ten snub polyhedra discovered by Coxeter and Miller in the early 1930's, but published only in 1954. Its equation is in Table 4 of Uniform polyhedra.

The measures listed on David McCooey's can be written in terms of and plastic constant . + The snub icosidodecadodecahedron of unit edge length element size circumscribed radius midscribed radius pentagon center radius triangle center radius pentagram center radius triangle-triangle angle pentagram-triangle angle pentagon-triangle angle With

McCooey's metrics panel gives volume , but no formula. However, (see Answer) is correct.

Graph The skeleton of a snub icosidodecadodecahedron can be represented as a graph with 60 vertices and 180 edges, an Eulerian graph.

Related polyhedra Medial hexagonal hexecontahedron

The medial hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.

See also Snubs Jonathan Bowers Uniform snubs Robert Ferréol Sided Richard Klitzing List of uniform polyhedra

References

Further reading

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