The stellated icosidodecahedron
We can look at the stellated icosidodecahedron in two different ways:
The stellated icosidodecahedron is simply the dodecahedron intersected with the icosahedron!
We can do this because the dodecahedron and the icosahedron have a very special relationship with each other: They are duals.
Dodecahedron: 12 faces, 20 vertices, 30 edges
Icosahedron: 20 faces, 12 vertices, 30 edges
They share many of the same properties -- They also share the same polyhedra when they are truncated. To see this, go back and take another look at the Icosidodeca Family.
The term "stellated" means that this polyhedron can be created by extending the faces of the icosidodecahedron. Can you see how the blue faces can be extended to form the blue dodecahedron on the left? The yellow faces will extend to form the yellow icosahedron.