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According to Wikipedia, groupoids is the appropriate tool for studying quasicrystals.

Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of the identical units of the crystal. The structure of crystals can be analyzed by defining an associated group. Quasicrystals, on the other hand, are composed of more than one type of unit, so, instead of lattices, quasilattices must be used. Instead of groups, groupoids, the mathematical generalization of groups in category theory, is the appropriate tool for studying quasicrystals. https://en.m.wikipedia.org/wiki/Quasicrystal#Mathematics

For a crystal, the associated group consists of all transformations under which the crystal is invariant.

My question is: for a quasicrystal, how is the associated groupoid defined? What are the elements (or objects) in the groupoid?

I did not find a specific explanation about the relationship between quasicrystals and groupoids, so I am confused.

Changsu Wang
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  • Can you provide an explicit link to that passage on Wikipedia? – Lee Mosher Oct 19 '22 at 14:10
  • The first think I would do is to read what more that Wikipedia entry has to say, and/or to follow any further links given in that entry. Did you try that? – Lee Mosher Oct 19 '22 at 14:10
  • link – Changsu Wang Oct 19 '22 at 15:01
  • The reference is not available, and I did not find a specific explanation about the relationship between quasicrystals and groupoids, so I am confused. – Changsu Wang Oct 19 '22 at 15:23
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    I think the first interpretation of qualicristas from the point of view of groupoids is due to Kellendonk (J. Kellendonk, Noncommutative geometry of tilings and gap labelling, Reviews in Mathematical Physics, 7(1995), 1133-1180). See also page 162 in Paterson's book "Groupoids, Inverse Semigroups, and their Operator Algebras". – Ruy Oct 19 '22 at 20:16

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