Inverted snub dodecadodecahedron

In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60.

[1] It is given a Schläfli symbol sr{5/3,5}.

be the largest real zero of the polynomial

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 snub dodecadodecahedron.

The edge length equals

, and the midradius equals

For a great snub icosidodecahedron whose edge length is 1, the circumradius is Its midradius is The other real root of P plays a similar role in the description of the Snub dodecadodecahedron The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron.

It is the dual of the uniform inverted snub dodecadodecahedron.

Its faces are irregular nonconvex pentagons, with one very acute angle.

Denote the golden ratio by

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

Then each face has three equal angles of

arccos ⁡ ( ξ ) ≈ 103.709

ξ + ϕ ) ≈ 3.990

Each face has one medium length edge, two short and two long ones.

, then the short edges have length

and the long edges have length

The dihedral angle equals

arccos ⁡ ( ξ

The other real zero of the polynomial

plays a similar role for the medial pentagonal hexecontahedron.

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