If you slice off a vertex of a dodecahedron, but not too deep, the slice leaves a new, triangular face. If you slice them all off to just the right depth so that the triangular faces meet each other halfway along the original edges (which thereby all disappear), you get this.
If you slice off a vertex of a icosahedron, but not too deep, the slice leaves a new, pentagonal face. If you slice them all off to just the right depth so that the pentagonal faces meet each other halfway along the original edges (which thereby all disappear), you get this.
Guess what? It's an icosidodecahedron!