Course MT3818 Topics in Geometry

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(The point groups)

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.

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 π about any lattice point is in Sd, but will in fact be bigger. It will, however, still be discrete.
Now let pM be the centre of a rotation R by 2π/n.
Let p1M be a closest point of M which is the centre of a rotation R1 by 2π/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 2π/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.


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 2π/4 while an equilateral lattice can have rotations by 2π/3 or 2π/6 as symmetries.


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(The point groups)

JOC February 2003