Lattices, An Introduction


This topic should not be confused with the (unrelated) definition of a lattice as a partially ordered set with meet and join.

By default, the "lattice points" in R3 are the points with integer coordinates. These form a regular grid throughout 3 space. The base cell is a cube, which is repeated again and again to fill the entire volume of space.

Stretch the points along the x axis, and the base cell becomes a rectangular box. Tilt the y axis and slant the lattice accordingly; the floor of each cell is now a parallelogram. Finally, tilt the z axis and each cell of the lattice is a parallelatope. It's basically a pushed over box.

After stretching and tilting, the lattice comprises the integer multiples of three vectors in 3 space. If these vectors are the coordinate unit vectors, the original cubic lattice appears.

If the three vectors lie in a plane, the lattice doesn't fill the entire space. Therefore the three vectors must be linearly independent.

Generalize this to n dimensions and you have the definition of a lattice in n space.

I like to think of a lattice as scaffolding that supports the surrounding space. Each small region is anchored by a lattice point.

Complex Space

A lattice can also be based on n linearly independent complex vectors in n dimensional complex space. The vectors are multiplied by complex integers, and added together.

Now picture the same lattice in real space, with twice as many dimensions. Each vector in Cn becomes a vector in R2n when multiplied by 1, and another vector in R2n when multiplied by i. If these vectors are linearly dependent over the reals, put the coefficients together, using i where necessary, and the original vectors are linearly dependent over the complex numbers. Therefore the new base vectors, in real space, are independent. Furthermore, the span of these real vectors, in real space, using integer coefficients, equals the span of the original complex vectors, in complex space, using complex integers as coefficients. The lattice is exactly the same; we have simply found a representation using real vectors in real space, with integer coefficients. This is often more convenient.