The rhombic icosahedron has 20 rhombic faces, ten of each of two different rhombs. The 5 faces which intersect at the axis of 5-fold symmetry have the same dihedral angle as a regular icosahedron.
The 20 rhombic faces are made from 22 balls and 40 struts whereas a regular icosahedron has 20 equalateral triangular faces made from 12 balls and 30 struts.
I also made a [stretched version](https://skfb.ly/6ysID) of this model.
This model was generated with [vZome](http://vzome.com). The download includes the vZome file.