In this book we will primarily be concerned with the properties and applications of self-avoiding polygons (SAP). Two closely related problems are those of polyomi-noes, and the much broader one of tilings. We will describe and discuss polyominoes and, in the context of a discussion of SAP, will briefly mention relevant aspects of the subject of tilings. In passing we will also concern ourselves with a discussion of some properties of polycubes. It will also turn out to be appropriate to discuss self-avoiding walks (SAW), as SAP can be usefully and simply related to a proper subset of SAW.
In all cases we shall be considering paths on a regular lattice. This will most often be the two-dimensional square lattice Z2, or its three-dimensional counterpart, Z3, called the simple-cubic (sc) lattice, or its higher dimensional analogue Z d , called the (d-dimensional) hyper-cubic lattice. Other two-dimensional lattices, such as the triangular (t) and hexagonal (h) lattices, and other three-dimensional lattices, such as the body-centred cubic (bcc), face-centred cubic (fcc) and tetrahedral or diamond (d) lattices will also be mentioned.