Visualizing Platonic Solid group symmetries

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How do you visualize the rotation symmetries, to classify a icosahedron for example as Ih, H3, [5,3], (*532)

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  • Face type: stick a rod through the midpoints of a pair of opposite faces and spin. Here is a bird's eye view.
  • Edge type: stick a rod through the midpoints of a pair of opposite edges and spin. It's harder to find a good bird's eye view of this one, but this image is close.
  • Vertex type: stick a rod through a pair of opposite vertices and spin. This image is a pretty good bird's eye view, if you get rid of the light bulb.

Added: You can keep track of the rotational symmetries by counting the number of opposite face pairs (there are 20 faces and 10 opposite face pairs), the number of opposite edge pairs (there are 30 edges and 15 opposite edge pairs), and the number of opposite vertex pairs (there are 12 vertices and 6 opposite vertex pairs).