The Solid phases can be ordered or amorphous. See the glassball simulation page for more details.
The ordered solids have many different lattice structures. Zero, one and two dimensional solids do exist, and even have technological applications but will not be discussed further here.
In three dimensions, there are 14 simple lattices called Bravais lattices. You can look at these here or on the almost completed site of Shahar Rozen . Sometimes the structures are more complex with several different sub-lattices. A common structure for semiconductors is the diamond structure.
Some elements can have different solid structures, e.g. Carbon which can take both diamond and graphite structures. See more details here and here. Additional symmmetries were believed for a long time not to exist, but two decades ago Dan Schectmann of Material Science at the Technion and collaborators found that a fivefold symmetry does exist, although it is not a completely longe range order. See more details here.
PHYSICS 3H - HOMEPAGE