The Groups of the Tetrahedron, the Octahedron, and the Icosahedron

icosahedron_smallAfter discussing the dihedral group, it is time to post my images of how Klein introduces the symmetry groups of the tetrahedron, the octahedron and the icosahedron.

By duality, this also handles the case of the dodecahedron (it is the dual of the icosahedron) and that of the cube (it is the dual of the octahedron) and thus all the Platonic solids are covered.

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The Dihedral Group

Snow flakes are one of nature’s beauties which are easy to appreciate even for the more mathematically or technically minded. Kenneth G. Libbrecht produces wonderful photographs of them; some are available online.

The symmetry of snow flakes is described by the dihedral group. This is one of the first groups described in Felix Klein’s book I have advertised before. Here is my illustration of how Klein described this group geometrically.

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Lectures on the Icosahedron

Some weeks ago, I was looking for examples giving me a quick overview on how to control transparency in the raytracer POV-Ray. This took me to the website of David Dumas who has beautiful illustrations of limit sets. (A very good and accessible introduction to the beauty of the Kleinian groups behind this is given by the book “Indra’s Pearls” [1].) What took me as even more beautiful were the pages from old mathematics books he is using as a background.

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