Course MT3818 Topics in Geometry

## Lattices

This is the study of discrete subgroups of *I*(**R**^{n}). 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 **R**^{2}.

**Definition**

A **lattice** *L* is a discrete subgroup of the group **R**^{n} (under addition).

**Remarks**

- We will usually assume that
*L* does not lie in a lower-dimensional subspace of **R**^{n}.

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

- We will usually think of
*L* as a subgroup of the group of translations in *I*(**R**^{n}).

One can get a basis for a lattice as follows.

Choose *e*_{1} to be the closest vector in *L* to **0**. Then all the vectors in *span*(*e*_{1}) are integer multiples of *e*_{1} .

Then choose a vector *e*_{2} not in *span*(*e*_{1}) but closest to it. Then every vector in *span*(*e*_{1} , *e*_{2}) is of the form *a*_{1}*e*_{1} + *a*_{2}*e*_{2} with *a*_{1} , *a*_{2} ∈ **Z**.

Continuing we get:
**Alternative definition**

A lattice *L* is the set {*a*_{1}*e*_{1} + *a*_{2}*e*_{2} + ... + *a*_{n}*e*_{n} | *a*_{i} ∈ **Z** } where {*e*_{1} , ... , *e*_{n} } is a basis of the vector space **R**^{n}.

**Remark**

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

**Examples**

- In
**R** the only lattice is essentially **Z**.

- A
*general* lattice in **R**^{2}

For this lattice we choose *e*_{1} and *e*_{2} with no special connections between the lengths or angles of the vectors.

- A
*rectangular* lattice in **R**^{2}

We may choose a basis in which *e*_{1} and *e*_{2} are perpendicular

- A
*square* lattice in **R**^{2}

A rectangular lattice in which ‖*e*_{1}‖ = ‖*e*_{2}‖

- A
*rhomboidal* or *isosceles* or *centred* lattice in **R**^{2
}

We have ‖*e*_{1}‖ = ‖*e*_{2}‖ but *e*_{1} and *e*_{2} are not necessarily perpendicular

- A
*hexagonal* or *equilateral* lattice in **R**^{2}

A lattice in which ‖*e*_{1}‖ = ‖*e*_{2}‖ and the angle beween *e*_{1} and *e*_{2} is π/3.

- A
*non-lattice* in **R**^{2}

- A
*simple cubic* lattice in **R**^{3}

- A
*body centred cubic* lattice in **R**^{3}

- A
*face centred cubic* lattice in **R**^{3}

Crystallographers classify lattices in **R**^{3} into 14 different types. They all appear in the structures of various chemical compounds.

JOC March 2003