Each of these dodecahedra were modified by truncations at exactly four of their three-valent vertices. As a result, each has four equilateral triangles as faces. In the one above, the Platonic dodecahedron’s pentagonal faces are modified into a dozen irregular hexagons by these truncations, while, in the one below, the rhombic dodecahedron’s faces are modified into twelve irregular pentagons.
Both of these polyhedra were created using Stella 4d, software you can try for yourself at this website.
The icosahedron has twenty triangular faces. Truncate it once, and the triangles become hexagons, with pentagons appearing under the pyramids removed in the truncation. This is the “soccer ball” shape familiar to millions.
If you take this figure and truncate it again, the twenty hexagons become twenty dodecagons, the twelve pentagons each become decagons, and sixty isosceles triangles appear under the pyramids removed by this second truncation.
I made this image using Stella 4d, a program you can find at www.software3d.com/Stella.php. Also, just for fun, here’s a version of it with the colors switched around, and with a slight bounce as it rotates in the other direction.
This is not the only truncated form of the rhombic dodecahedron. In this polyhedron, square-based pyramids have been “truncated away” from the rhombic dodecahedron’s four-valent vertices, but the three-valent vertices remain untouched.
If this truncation is done in such a way as to leave the hexagonal faces equilateral (which is not done here), so that all edges of the polyhedron have the same length, the result is called a “chamfered cube.” This closely-related figure may be seen at https://en.wikipedia.org/wiki/Chamfered_cube.
[Image credit: this rotating model was created using Stella 4d, software you can find right here, with a free trial download available.]