Talk:Crystallographic restriction theorem

message: 4D examples with isometries of order 5, 8, 10, and 12 would be interesting.--Patrick 10:47, 8 November 2005 (UTC)
thread_title:  4D examples 

message: I tried to read carefully the article and the talk page. I suggest a compromise. I do like Patrick's formulation, but I think it is kind of intimidating especially given that everything else in this article is rather elementary. I tried to thus put it at the bottom as a higher-math perspective. 
In this way no content is lost, and people can get the more abstract perspective, but not at the expense of sacrificing a good read for people like me who have no idea of what a discrete isometry is. By the way, I am not an expert, so if the compromise sticks, please do edit that new section. I tried to convey in there some motivation for the isometries formulation, some more might be needed. Oleg Alexandrov (talk) 00:57, 9 November 2005 (UTC)
thread_title:  Compromise? 

message: I rephrased the introduction to make it more precise both for the novice and the expert reader.  For one thing, I did not like the caveat that the theorem doesn't have to apply to finite crystals.  Yes, all crystals are finite, but they are plenty big enough for the theorem to hold.
Greg Kuperberg 17:32, 14 February 2007 (UTC)
replies: text: Reverted. And your statement that all crystals are "big enough" is mathematical nonsense. The physical observation of crystal symmetry is different from the mathematical proof. A symmetry group that includes translation must be infinite. Quantum chemistry uses the tools of mathematics for explanations, but the universe does what it pleases. When we discover quasicrystals, we do not falsify any theorems. --KSmrqT 00:04, 15 February 2007 (UTC)
thread_title:  Clearer introduction 

message: I undid KSmrq's revert of my "unhelpful edits"; here is why:
* 1: I changed the intro because it was misleading: it falsely implied that quasiperiodic structures have to be finite "in the mathematical formalism" (which is nonsense, because a crystal is not a "mathematical formalism"). A Penrose tiling, for example, is both infinite and quasiperiodic.
* 2: I removed the "compatible" part of the proof. It is not actually necessary, and as far as I can tell, it does not appear in Scherrer's paper. But I don't know German, so I could be mistaken in this.
* 3: I replaced the not-very-relevant example of a 4d double rotation with a better one, which actually shows an eightfold symmetry of the lattice. 
Ben Standeven (talk) 19:25, 17 September 2008 (UTC)
thread_title:  The reasoning behind my changes. 

message: Can anyone explain what is meant by "Due to periodicity of the crystal, the new vector r' which connects them must be equal to an integral multiple of r:"? Why should the vector A'B' be along any of the basis directions at all? Here is a case where I see a problem - take the hexagonal lattice, take A = (0,1), B = (0,2) and R = rotation by 60 degrees about (0,0) - then A' = (-1,-1), B' = (2,0), and A'B' is not a vector along the basis directions. Here the coordinates are w.r.t the primitive cell. Am I missing anything in the proof?  —Preceding unsigned comment added by 210.212.55.3 (talk) 10:50, 8 October 2009 (UTC)
thread_title:  A question about the second proof 

message: The trigonometry proof used to let the point A be rotated in the opposite direction to the point B, which seemed right. Unless both rotations are in the same plane, and one is clockwise while the other is counter-clockwise (by -\alpha), A'B' doesn't end up parallel to AB for most rotation angles. The "opposite direction" phrase got removed in a recent edit, that also includes some other changes however. Wrwrwr (talk) 17:00, 5 January 2015 (UTC)
thread_title:  Trigonometry proof: rotations direction