Course MT3818 Topics in Geometry

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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 , a2 belongs Z.
Continuing we get:

Alternative definition
A lattice L is the set {a1e1 + a2e2 + ... + anen | ai belongs Z } 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 norme1norm = norme2norm


  5. A rhomboidal or isosceles or centred lattice in R2

         

    We have norme1norm = norme2norm but e1 and e2 are not necessarily perpendicular


  6. A hexagonal or equilateral lattice in R2
    A lattice in which norme1norm = norme2norm and the angle beween e1 and e2 is p/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.


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(Contents of advanced course)
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(The crystallographic restriction)

JOC March 2003