Deltoidal hexecontahedron

The other two angles of each face, between a short and a long edge each, are both equal to arccos(⁠5-2√5/10⁠)≈86.97415549104°.

The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.

The 62 vertices of the deltoidal hexecontahedron fall in three sets centered on the origin: These hulls are visualized in the figure below: The deltoidal hexecontahedron has 3 symmetry positions located on the 3 types of vertices: The deltoidal hexecontahedron can be constructed from either the regular icosahedron or regular dodecahedron by adding vertices mid-edge, and mid-face, and creating new edges from each edge center to the face centers.

Conway polyhedron notation would give these as oI, and oD, ortho-icosahedron, and ortho-dodecahedron.

These geometric variations exist as a continuum along one degree of freedom.