It has long been recognised that this evidence of external regularity is related to the regularity of internal structure. Diffraction techniques are now available which give much more information about the internal structure of crystals, and it is recognised that internal order can exist with no external evidence for it. Crystallography Repeating patterns can be described in terms of their symmetry. Similarly, the regularity of the internal structure of crystals can be described formally in terms of symmetry elements. 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.