**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: