Medial rhombic triacontahedron

In geometry, the medial rhombic triacontahedron (or midly rhombic triacontahedron) is a nonconvex isohedral polyhedron.

It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron.

This means that on each axis there is an inner and an outer vertex.

The ratio of outer to inner vertex radius is

It has 30 intersecting rhombic faces, which correspond to the faces of the convex rhombic triacontahedron.

The diagonals in the rhombs of the convex solid have a ratio of 1 to

The medial solid can be generated from the convex one by stretching the shorter diagonal from length 1 to

So the ratio of rhomb diagonals in the medial solid is 1 to

This solid is to the compound of small stellated dodecahedron and great dodecahedron what the convex one is to the compound of dodecahedron and icosahedron: The crossing edges in the dual compound are the diagonals of the rhombs.

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

It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares.

As such, it is topologically a regular polyhedron of index two:[1] Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.