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

ina
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1 Answers1

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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).

Lee Mosher
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  • Does this only work for polyhedra with opposite faces/edges/vertices? – ina Aug 23 '14 at 17:14
  • what are some ways to keep track of each configuration? – ina Aug 23 '14 at 17:17
  • @Ina: For your first question, it works exactly as stated for all of the Platonic solids except for the tetrahedron. For your second question, I will add some stuff to my answer. – Lee Mosher Aug 23 '14 at 17:23
  • How does it work for the tetrahedron? – ina Aug 23 '14 at 17:29
  • @Ina: Edge pairs are the same, six edges and three opposite edge-pairs. But the four vertices and four faces are paired up into four opposite-vertex-face pairs. – Lee Mosher Aug 23 '14 at 17:37
  • I'm still not sure how the symmetry visualization for the tetrahedron works in this poking-nomenclature.. – ina Aug 24 '14 at 14:21
  • also! how do you diagram these? http://math.stackexchange.com/questions/907734/reading-the-sphere-diagrams-in-point-groups-on-wikipedia – ina Aug 24 '14 at 14:22