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My teacher Kirsi of Mat-1.3000 in Aalto University stated 17 symmetry classes for planes and 14 for spherical things (some lecture slides here). She used Conway Thurston's notation to classify symmetries. Now what is the amount of symmetry classes with cylinders or what about torus or other shapes? General topological idea?

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I am trying to find out some literature to understand this topic deeper about orbifold notation and the classes. My goal is to learn this by classifying everyday things such as tires with repeating patterns. References welcome.

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First of all, a small correction: it's Conway-Thurston notation, named after two separate people: John Horton Conway and William Thurston. Secondly, their notation (in terms of point symmetries) actually handles the cylinder, in the form of frieze patterns: these are essentially just 'infinite unrollings' of a cylinder. As for the torus, that's also handled — all of the symmetries of the plane are (essentially) symmetries of a torus! This is the case because the plane is the unrolling of a torus in the same way as the infinite strip is the unrolling of a cylinder. (This theorem — the idea that every symmetry pattern of the plane has a rectangular 'fundamental region' that can be used to reproduce the pattern — is a nontrivial one in its own right.)

Incidentally, I'm considering the torus here as an abstract entity, that is, a fundamentally flat surface that happens to be periodically symmetric in both directions; if you're referring to a physical torus then the possible symmetries are much smaller, because the group of motions that map the torus onto itself is much smaller - it consists of only rotations about the torus's central axis and certain reflections and 'flips' about planes or lines through the origin, and all of these are easily classified as subgroups of the sphere's symmetry groups (they amount to the various 'N' symmetries of the sphere, I believe).

If you're interested in more details, both about the notation and about the classification of patterns, you can't do any better than the recent amazing book on the subject, The Symmetries of Things. I don't want to give a full review here, but my guess is that you'll find both material you can understand and even more material the book will make you want to understand; it is IMHO the perfect introduction to the subject, an instant classic.

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I cannot fully understand how torus could be a subgroup of sphere: you could tie the inside so that it looks like a sphere but it is still a torus -- it still has a hole while sphere not. Thank you for the book recommendation +1, trying to get my hands to it before further ado :) – hhh Feb 9 '13 at 0:51
@hhh The simplest way to see it might be this: consider all of the isometries - rigid motions of space - that leave the torus invariant (that is to say, that keep the surface of the torus occupying exactly the same space that it was initially); can you see why all of these would also leave a sphere centered at the torus's center invariant? – Steven Stadnicki Feb 9 '13 at 0:56

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