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The crystallography restriction states that any 2-dimensional lattice can have rotational symmetry of degree 1, 2, 3, 4 or 6 - and that's it. A simple proof of I've heard of this is: the magnitude of $rot_\theta(x)-x$ is less than the magnitude of $x$ if $\theta$ is less than $\pi / 3$. (So if there's a rotation by less than $\pi/3$, it will fail to be a lattice, since we can always get smaller and smaller elements.)

That sounds fine to me, and I get why real crystals would have the requirement that all atoms be some epsilon away from each other. I don't understand why nature requires the lattice to be closed though - obviously real crystals aren't completely closed under addition, or else they would be infinitely large.

So why does the crystallography restriction limit the symmetries of real crystals? (Or does it not, and this application of the theorem is made up by math teachers to convince their students that math is useful?)

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I'm sorry - I'm failing to understand your question. Would you please clarify? The proof of this is fairly trivial and found in any CM text, but from your question, you indicate that you understand translational invariance, but not the restriction on symmetry, even though this is provided in the wikipedia article, so is your question really one of the application of group theory to crystallography? – Jen Sep 24 '11 at 15:04
@Jen: Why is it true that if x and y are in a crystal, x+y is too? (i.e. why are crystals closed?) – Xodarap Sep 25 '11 at 3:08
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It really is just a definition, as Misha said. A crystal is defined as a state of matter where translational invariance is broken to a discrete subgroup. It occurs when you have a simple molecule substance, where units are rigidly arranged in a packing. The densest packings of most non-contrived simple shapes repeat with a particular periodicity.

You can arrange for complex molecules to crystallize, but it requires special preparation in aquaous solution, and proper ionic concentrations, which surround the complex molecule in a cage of a certain type. If you get a protein to crystallize, you write a paper, because then you can take an atomic scale picture of it using X-ray diffraction.

If you just jam proteins together and cool them down you make amorphous non-translational things, like gels or rubbers. These are heavily tangled polymer chains with no translational subgroup preserved. You don't study them, because they are complicated. They have a ground state entropy, which is caused by the trapping of the gel in metastable states. These systems are the subject of active research, and their theory is not well understood. Whether glasses have a sharp liquification transition was debated for decades (current consensus is that they do).

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Thanks Ron. So roughly: the structure is periodic because that's the densest packing possible? – Xodarap Sep 27 '11 at 14:16
@Xodarap: yes-sirree-bob ( character limits forbid a simple yes). – Ron Maimon Sep 27 '11 at 18:48

Crystal lattice has translational invariance by definition. Period.

The thing you are interested in has no translational invariance. Thus, it is not what people usually call crystal lattice. You might be interested in reading about quasicrystals. Which are more or less like crystal, but have a bit relaxed approach to periodivity.

By the way, if you count the number of atoms in real crystal you might get an idea why it is not that far from infinity.

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So what is called the "crystallography restriction" does not restrict the shape of physical crystals? – Xodarap Sep 25 '11 at 15:40
It depends on what you call "physical crystals". Conventional definition, as I mentioned, does restrict. Could you explain in more details, what you have against translational invariace? Crystals are large, surface is small. If you want to forget the surface, you should assume that they are infinite. Which is pretty close to true taking into account that the size of the real crystal almost infinitely larger that atomic size. – Misha Sep 25 '11 at 17:05
I guess my question might be better phrased as: why do some solids exhibit translational invariance? It doesn't appear to me that all solids have this property, so there must be something to separate crystals from non-crystals. – Xodarap Sep 27 '11 at 0:16
It was kind-of-approved in the beginning of XX century by X-ray diffraction. More or less the same argument distinguish solids with and without translational invariance. If a piece of material shows good X-ray diffracion picture, which is equivalent to the fact that Fourier transform of atomic coordinates is well approximation of sum of delta-functions, this means that in some sense translational invariance is a good approximation. – Misha Sep 27 '11 at 4:16

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