# Rotational Symmetries of the Dodecahedron

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.

• Shouldn't $4\times12$ be $4\times6$? There are $12$ faces so $6$ pairs of opposite faces, right? – bof Oct 12 '18 at 1:13
• @bof It is amazing how hard counting is. Thanks! – Andrew Shedlock Oct 12 '18 at 1:16

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.