Great inverted snub icosidodecahedron

In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69.

It is given a Schläfli symbol sr{5⁄3,3}, and Coxeter-Dynkin diagram .

In the book Polyhedron Models by Magnus Wenninger, the polyhedron is misnamed great snub icosidodecahedron, and vice versa.

be the largest (least negative) negative zero of the polynomial

is the golden ratio.

is the rotation around the axis

Let the linear transformations

be the transformations which send a point

with an even number of minus signs.

constitute the group of rotational symmetries of a regular tetrahedron.

constitute the group of rotational symmetries of a regular icosahedron.

are the vertices of a great snub icosahedron.

The edge length equals

, the circumradius equals

, and the midradius equals

For a great snub icosidodecahedron whose edge length is 1, the circumradius is Its midradius is The four positive real roots of the sextic in R2,

are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57), great inverted snub icosidodecahedron (U69), and great retrosnub icosidodecahedron (U74).

The great inverted pentagonal hexecontahedron (or petaloidal trisicosahedron) is a nonconvex isohedral polyhedron.

It is composed of 60 concave pentagonal faces, 150 edges and 92 vertices.

It is the dual of the uniform great inverted snub icosidodecahedron.

Denote the golden ratio by

be the smallest positive zero of the polynomial

Then each pentagonal face has four equal angles of

Each face has three long and two short edges.

between the lengths of the long and the short edges is given by The dihedral angle equals

Part of each face lies inside the solid, hence is invisible in solid models.

The other two zeroes of the polynomial

play a similar role in the description of the great pentagonal hexecontahedron and the great pentagrammic hexecontahedron.

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