Crystal Systems
There are seven different crystal systems. Each of these is defined by certain minimum symmetries which lead to the adoption of certain lattice types compatible with the symmetry of the crystal system. It is perhaps useful to consider a causal relationship between the symmetry of a crystal and the lattice adopted by that crystal. If a crystal contains a four-fold rotation axis, the lattice adopted by the crystal has certain restrictions placed on its axes and angles. This concept will be developed as the course is progressed. The table below shows the required symmetries for each crystal system.
| Crystal System | Required Minimum Symmetry |
| Triclinic | None |
| Monoclinic | Mirror plane or two-fold rotation axis |
| Orthorhombic | 3 x two-fold rotation axes or 1 x two-fold rotation axis and 2 x mirror planes |
| Tetragonal | Four-fold rotation axis |
| Trigonal | Three-fold rotation axis |
| Hexagonal | Six-fold rotation axis |
| Cubic | 4 x three-fold rotation axes |
As seen in the table above, two, three, four and six-fold rotations are compatible with periodic crystals. A simple explanation of ‘regular pentagons and regular shapes with 7 or more sides do not tessellate’ is often given for this. Crystals with five-fold rotational symmetry are possible, but these lack translational symmetry and are thus known as ‘quasicrystals‘. For those more interested in the maths, the Wikipedia page on the Crystallographic Restriction Theorem provides multiple useful proofs on why certain rotation axes are incompatible with periodic crystals.