The icosahedron has to be the most neglected Platonic solid. The cube and tetrahedron are the most familiar. You have probably seen the dodecahedron with a calendar on it. An octahedron is just two Egyptian pyramids joined together at their bases.

With 20 faces and 30 edges, an icosahedron is quite an eyeful. I found this site to help visualize it. They could have done a better job of coloring, but if you stop the spinning and choose the beam option, it is not too hard to see what is going on.

Manipulate the shape so there is a top vertex and a bottom vertex,
each attached to 5 triangles. That accounts for 10 faces. Then there is a middle row, with 5 triangles attached at their base to the top row alternating with 5 triangles attached at their base to the bottom row. That makes for a total of 20.

Counting the edges can now also be done. The top vertex is attached to 5 edges. Each of the 5 triangles formed has a base edge. That is a total of 10 edges. The bottom vertex gives another 10 edges. The middle row has extra edges separating each of the 10 triangles. That adds up to 30.