This is the rhombicosidodecahedron, one of the thirteen Archimedean solids.
Several of the 92 Johnson solids are modified forms of this polyhedron, such as J76, the diminished rhombicosidodecahedron (shown below). It is formed by removal of a pentagonal cupola from a rhombicosidodecahedron, exposing a decagonal face.
Another variant of this Archimedean solid may be created by removing two pentagonal cupolas, exposing decagons on opposite sides of the figure. This solid, J80, is called the parabidiminished rhombicosidodecahedron.
Two J76s and one J80 can then be joined together, at their decagonal faces, to form this: the triple rhombicosidodecahedron.
I made these using Stella 4d, a program you can try for free at this website.