Repeating patterns, unit cells, and lattices

A repeating pattern is formed by the repetition of some unit at regular intervals along one, two, or three (non-parallel) axes which, with the repeat distances, define the lattice on which the pattern is based. For a 1-dimensional pattern the lattice is a line, for a 2-dimensional pattern it is a plane network, and for a 3-dimensional pattern a third axis is introduced which is not coplanar with the first two (Fig. 2.1). 

The parameters required to define the three types of lattice in their most general forms are 

These parameters define the unit cell (repeat unit) of the pattern, which is accentuated in Fig. 2.1. Any point (or line) placed in one unit cell must occupy the same relative position in every unit cell, and therefore any pattern, whether 1-, 2-, or 3-dimensional, is completely described if the contents of one unit cell are specified. 

Clearly, every lattice point has the same environment. A further property is that if along any line in the lattice there are lattice points distance x apart then there must be points at this separation (and no other lattice points) along this line when produced indefinitely in either direction. (We refer to this property of a lattice when we describe the closest packing of spheres in Chapter 4.) Note that the lattice has no physical reality it does not form part of the pattern. 

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