# Two Dimensions and Beyond

## Introduction

The physical world in which we live involves three dimensions of space. Yet so far in this book all the systems we have discussed have effectively been limited to just one dimension.

The purpose of this chapter, therefore, is to see how much of a difference it makes to allow more than one dimension.

At least in simple cases, the basic idea—as illustrated in the pictures below—is to consider systems whose elements do not just lie along a one-dimensional line, but instead are arranged for example on a two-dimensional grid.

Examples of simple arrangements of elements in one, two and three dimensions. In two dimensions, what is shown is a square grid; triangular and hexagonal grids are also possible. In three dimensions, what is shown is a cubic lattice; various other lattices, analogous to those for regular crystals, are also possible—as are arrangements that are not repetitive.