Course MT3818 Topics in Geometry

 Previous page (Contents of advanced course) Contents Next page (The crystallographic restriction)

## Lattices

This is the study of discrete subgroups of I(Rn). We shall mainly look at the case n = 2.
Our aim is to classify the two-dimensional crystallographic groups in much the same way as we classified the Frieze groups.

In the case of the frieze groups, we took a "smallest translation" and looked at what happened when we applied this to a point. (This is called the orbit of the point.) This gave us a subset of R which we might as well take to be the additive subgroup Z.
We now do something rather similar in R2.

Definition
A lattice L is a discrete subgroup of the group Rn (under addition).

Remarks

1. We will usually assume that L does not lie in a lower-dimensional subspace of Rn.

2. Recall that discrete means that there is a minimum distance between points of L.

3. We will usually think of L as a subgroup of the group of translations in I(Rn).

One can get a basis for a lattice as follows.
Choose e1 to be the closest vector in L to 0. Then all the vectors in span(e1) are integer multiples of e1 .
Then choose a vector e2 not in span(e1) but closest to it. Then every vector in span(e1 , e2) is of the form a1e1 + a2e2 with a1 , a2Z.
Continuing we get:

Alternative definition
A lattice L is the set {a1e1 + a2e2 + ... + anen | aiZ } where {e1 , ... , en } is a basis of the vector space Rn.

Remark
Note that the basis constructed above is not the only possible basis.

Examples

1. In R the only lattice is essentially Z.

2. A general lattice in R2
For this lattice we choose e1 and e2 with no special connections between the lengths or angles of the vectors.

3. A rectangular lattice in R2
We may choose a basis in which e1 and e2 are perpendicular

4. A square lattice in R2
A rectangular lattice in which ‖e1‖ = ‖e2

5. A rhomboidal or isosceles or centred lattice in R2  We have ‖e1‖ = ‖e2‖ but e1 and e2 are not necessarily perpendicular

6. A hexagonal or equilateral lattice in R2
A lattice in which ‖e1‖ = ‖e2‖ and the angle beween e1 and e2 is π/3.

7. A non-lattice in R2

8. A simple cubic lattice in R3

9. A body centred cubic lattice in R3

10. A face centred cubic lattice in R3

Crystallographers classify lattices in R3 into 14 different types. They all appear in the structures of various chemical compounds.

 Previous page (Contents of advanced course) Contents Next page (The crystallographic restriction)

JOC March 2003