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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.

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    $\begingroup$ Shouldn't $4\times12$ be $4\times6$? There are $12$ faces so $6$ pairs of opposite faces, right? $\endgroup$ – bof Oct 12 '18 at 1:13
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    $\begingroup$ @bof It is amazing how hard counting is. Thanks! $\endgroup$ – Andrew Shedlock Oct 12 '18 at 1:16
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For "$4\times12=48$" read $4\times6=24$, since there are $6$ pairs of opposite faces. With this correction you have $1+15+24+260$ rotations, the right number.

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