I will demonstrate why the group of rotational symmetries of a Dodecahedron is $A_5$. For that, we have to find five nice objects, on which the group of symmetries acts.
One object is "Cubes" inscribed in Dodecahedron. But since my demonstration will be on "Black-Board" rather than "Slide-Show". Can anybody suggest me a simple way to describe five cubes in Dodecahedron, which are the principal objects of permutations in the group of rotations symmetries of Dodecahedron?