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On an old paper, I found an interesting representation of some symmetry operations: stereographs.

Online I found a set of symmetry operations and related stereographs (see: http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Stereographs.html), yet it's not clear to me how the representation is related to the symmetry operations (or: how are the representations obtained?).

Some samples from the website linked above:

  • C4

    C4

  • C5h

    C5h

  • D6

    D6

I would like to understand the geometrical (and symbolical) meaning of a stereograph, in order to relate non-mnemonically it to its symmetry operation

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    $\begingroup$ Just to be clear: You are asking about the difference between things like $\oplus$ and $+$? $\endgroup$ – Feodoran Nov 12 '18 at 18:39
  • $\begingroup$ Not only that. Apparently, although I couldn't find a complete description, stereographs are obtained via "mathematical" projection over a sphere of points derived from the symmetry of the object. Then, the location of the points gives the symbol in the stereographs. I think that every strange symbol in a stereograph could be understood in those terms. I am sorry if the description is terrible, but I couldn't really find anything "complete" about this notation. If someone could suggest, for lack of a description, a book which explains it, I could make a search in my university's library $\endgroup$ – The_Vinz Nov 12 '18 at 20:01
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    $\begingroup$ I vaguely remember a book with all the space groups (regarding symmetry in solid state chemistry), which had similar pictures. I just don't remember its name. But maybe this can be of help: img.chem.ucl.ac.uk/sgp/misc/symbols.htm $\endgroup$ – Feodoran Nov 12 '18 at 20:07
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    $\begingroup$ The books ' The Basics of Crystallography and Diffraction' by C. Hammond, IUCR texts on Crystallography, publ OUP, and McKie & McKie, 'Essentials of Crystallography, publ Blackwell Scientific, both contain chapters on these projections. $\endgroup$ – porphyrin Nov 13 '18 at 8:31

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