When I did Icosidodecahedron curler unit, I went curious about this polyhedron and searched on the net. I found some really interesting facts I thought worth documenting here for quick reference.
Archimedean solid, (From Wiki) it is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices. They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices.
Icosidodecahedron and Cuboctahedron are one of the 13 Archimedean solids. For more refer to this link: http://www.geom.uiuc.edu/~sudzi/polyhedra/archimedean.html
There are 5 (and only five) Platonic solids namely Cube, Tetrahedron, Octahedron, Dodecahedron and Icosahedron.
Truncation means cutting off the corners of a solid. We cut off identical lengths along each edge emerging from a vertex. This process adds a new face to the polyhedron.
What happens when we truncate Icosahedron?
If, when truncating the vertices of either the icosahedron or the dodecahedron, we move in exactly half way, the result is the icosidodecahedron. Thus, we can think of the icosidodecahedron as being the limiting case of either the truncated icosahedron or the truncated dodecahedron.
There's another Archimedean solid worth mentioning. That is Truncated Icosahedron. (polyhedron in Ten intersecting Planes)While truncating Icosahedron if we move 1/3rd we get Truncated Icosahedron. A good example of this solid is a Football.
And if we move 1/2 we get Icosidodecahedron.
There's a reason why we arrive at Icosidodecahedron when we truncate either Icosahedron or Dodecahedron half way. The reason is they are Dual Solids where the vertices of one corresponds to faces of other.
If we take Platonic Solids:
1) Dodecahedron and Icosahedron are dual solids.
Explanation: Dodecahedron has 12 faces and 20 vertices
Icosahedron has 20 faces and 12 vertices
2) Cube and Octahedron are dual solids.
Explanation: Cube has 6 faces and 8 vertices
Octahedron has 8 faces and 6 vertices.
3) Tetrahedron is self dual. It has 4 faces and 4 vertices.
Refer to my Origami platonic pictures to verify the above fact.
http://www.geom.uiuc.edu/~sudzi/polyhedra/ - In this site, under Archimedean solids, there's lovely illustration of diagrams to help understand better.