Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there consequences of the unsolvability of $A_5$ that can be viewed in terms of the geometry of the icosahedron?
I am seeking some connection for pedagogical reasons, from the geometry to the group theory. One could argue from orbits of vertices of the icosahedron under its various symmetries that $A_5$ has no (nontrivial) normal subgroups, but it is a long way from there to the unsolvability of $A_5$ for students who have not yet studied group theory.
Thanks for any connections you can suggest!
Update 1. Answered by Barry Cipra and Geral Edgar:
No longer $2.25, but still inexpensive through used-booksellers.
Update 2. And here is Jerry Shurman's book (PDF download), as cited by Barry and Sam Hopkins: