In this section we consider the symmetry and the categorisation of patterns and crystals in just two dimensions to simplify the ideas presented. Reflection symmetry is demonstrated and then rotation symmetry is explained. Exercise pages allow the user to identify reflection and rotation symmetry in 2-D shapes. The combination of reflection and rotation symmetry to give the ten plane point symmetry groups follows. Two dimensional patterns and crystals can be built up from a single unit of pattern called a basis, by repeated translations, due to two vector operators, known as unit vectors.
Unit vectors outline a repeated area of a pattern called a unit cell.
Repeated translations of unit vectors mark out a grid of identical points in space, called a lattice. The concept that patterns and crystals can be built by associating an identical basis in the same way with each point of a lattice is demonstrated. Unit cells can be divided into primitive and non-primitive types. All 2-D lattices can be categorised into one of five classes of plane lattices. The unit cells for each of these five lattices are shown.