I have been trying the build up my intuition on finding rotational symmetries of shapes and I have been looking at the dodecahedron from the platonic solids. I am convinced that I have the correct number of rotational symmetries, but I should only be finding $60$ rotational symmetries and $120$ if we include inversion symmetries.
The rotational symmetries I have found include $1$ identity. If we look at the opposite edges and draw a line which bisects both of these opposite edges then we find a $180$ rotation for every pair of opposite edges, which introduces $15$ new axis of rotations. Next we have an axis going through every pair of opposite faces and have 4 non-trivial rotations for every pair of opposite faces. This adds $4\times12 = 48$ new rotations. Lastly I looked at the opposite pairs of vertices and we may rotate the axis which passes through both vertices and have $2$ more rotations for each pair of vertices. This adds $20$ more rotations.
With this I think I have found $1 + 15 + 48 + 20 = 84$ rotational symmetries, which looks like I am over counting somewhere but I am not sure where.
Best Answer
For "$4\times12=48$" read $4\times6=24$, since there are $6$ pairs of opposite faces. With this correction you have $1 + 15 + 24 + 20 = 60$ rotations, the right number.