Course MT3818 Topics in Geometry

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The crystallographic restriction

We now look at which groups can be the symmetry groups of lattices.

Note that if L is the lattice of translations of a symmetry group G then L is a normal subgroup of G and the quotient group G/L acts on L by conjugation.

The main result is:

The Crystallographic restriction
Any rotation in the symmetry group of a lattice can only have order 2, 3, 4, or 6.

Proof
We will give the proof for R2. The proof for R3 is similar. It is harder for higher dimensions!
Let L be the lattice and let M be the set of all centres of rotations in Sd(L). This will include L since rotation by p about any lattice point is in Sd, but will in fact be bigger. It will, however, still be discrete.
Now let p belongs M be the centre of a rotation R by 2p/n.
Let p1 belongs M be a closest point of M which is the centre of a rotation R1 by 2p/n.
Let p2 = R1(p).
Now if T is any transformation mapping a point x to T(x) then conjugating a rotation about x by T gives a rotation (by the same angle) about T(x).
Thus conjugating R by R1 gives a rotation R2 by 2p/n about the point p2 and the diagram shows that if n > 6 the point p2 would be closer to p than p1 contradicting the definition of p1 .


A similar proof using this diagram:
with p3 = R2(p1), rules out the possibility that n = 5.




Remark

Note that the cases n = 2 , 3 , 4 and 6 are all possible.
A general lattice can have half-turns as symmetries, a square lattice can be left fixed by rotations by 2p/4 while an equilateral lattice can have rotations by 2p/3 or 2p/6 as symmetries.

     


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(Lattices)
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JOC February 2003