The periodic internal structure of minerals causes minerals to occur with regular external forms - as crystals. The angles between two faces of a crystal is identical for all forms of the same mineral that exhibit the corresponding faces. This is true regardless of the size or place of origin of the crystals! The consistency of interfacial angles was first reported by Nicolas Steno in 1669.
Interfacial angles for well-developed specimens of mineral crystals are measured perpendicular to the line of intersection of two crystal faces. The angle reported is always the acute angle. Interfacial angles can be measured with a contact goniometer, or more precisely with a reflection goniometer.
The observation of the consistency of interfacial angles led to the suggestion that crystals are built according to a pattern from building blocks that are small relative to the size of a macroscopic crystal. The concept of these building blocks is essentially the same as that of the unit cell in modern crystallography.
This pattern, or ordered atomic arrangement, implies that a certain atom is present in exactly the same structural (atomic) site throughout an infinite atomic array. To say that an atom is in the "same atomic site" means that it is surrounded by an identical arrangement of neighboring atoms.
To create a pattern requires a repeating unit or motif and instructions for how to perform the repeat. The recipe for a specific pattern may be discovered from the pattern by making some careful observations:
Note that the lattice can be thought of as an imaginary pattern of points (or nodes) in which every point has an environment that is identical to that of any other point in the pattern. A lattice has no specific origin, as it can be shifted parallel to itself. The lines along which the lattice can be shifted , i.e., translated, are the lattice vectors.
The various symmetry elements discussed thus far, with the addition of glide planes (translation), leads to formation of 17 Plane Groups, as shown on page 121 in K&H.
One way to consider the symmetry of lattices is to examine patterns or works of art that are based on repeating motifs, tiles, or tesselations. Work of the Dutch artist M.C. Escher contains many such examples. Click on the figures below to see color versions of Escher's work (each file is about 50 k), and explore the topic more through the 'World of Escher' or by searching Escher with your browser. There is even a demo version of a program called TesselMania!, for generating your own tesselations, that you can download and try.
For any given lattice, there are many possible choices of unit cell. Normally, the unit cell translations are selected so that there is one unit cell for each lattice point (i.e., for an infinite lattice, the number of unit cells equals the number of lattice points). Any such unit cell would be a primitive unit cell. All primitive unit cells have the same area (or volume). In some instances, however, it is convenient to choose a larger unit cell such that there is one unit cell for every two (or more) lattice points. These are called centered unit cells. Centered unit cells are usually chosen to have a rectangular geometry.
Unit cell translations may be used to define a coordinate system for a crystal. Points are located in crystallographic coordinate systems by measuring parallel to the unit cell translations.
Unit cell translations and the axes of crystal coordinate systems are labeled a, b, and c. Their positive directions are selected to define a right-handed coordinate system: to the right and front and upward are positive directions, to the left and back and down are negative. The angles between unit cell translations are identified by the Greek letter corresponding to the name of the opposite (i.e., perpendicular) translation. (Note, my html editor won't do no Greek!) Gamma is the angle opposite c (i.e. the angle between positive a and positive b). Similarly, the angles opposite a and b are alpha and beta, respectively.
