Since Prof. McKenzie brought it up, I thought it would be sensible to post about Bravais lattices and space groups (the 230 different symmetry groups allowed with the 14 Bravais lattices). Essentially, a space group is a particular arrangement of the elements in a unit cell which has a certain number of symmetry elements.
This description does not give us any information about the stability of these space groups, so it turns out that some groups are overwhelmingly common in nature and others have only a few examples (I'll have to check my notes to remember which ones are rarest).
Another interesting (but very useful fact) is that proteins (which are chiral in nature, i.e. have a handedness) cannot only form crystals with space groups that have mirror plane symmetry unless the other conformation is present also. Since biological proteins generally only occur with one handedness, it's impossible for them to form these crystal symmetries. This cuts down on the possible numbers of space groups that are allowed and thus makes x-ray crystallography of proteins much easier (but still not easy).
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This highlights the usefulness of symmetry to simplify complicated problems. Even in quantum, there are tricks using symmetry properties of Hamiltonian to find the eigenvalues and eigenfunctions easily for problems with a large Hilbert space.
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