Symmetry in three dimensions

This chapter follows, in outline, the format of Chapter 3. In this chapter, though, three-dimensional crystals form the subject matter, and the two-dimensional concepts already presented will be taken into this higher dimension. 

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Strictly speaking, a symmetry-translation is only possible for an infinitely extended object. An ideal crystal is infinitely large and has translational symmetry in three dimensions. To characterize its translational symmetry, three non-coplanar translation vectors a, b and c are required. A real crystal can be regarded as a finite section of an ideal crystal this is an excellent way to describe the actual conditions. 

Several kinds of intermediate states exist between the state of highest order in a crystal having translational symmetry in three dimensions and the disordered distribution of particles in a liquid. Liquid crystals are closest to the liquid state. They behave macroscopically like liquids, their molecules are in constant motion, but to a certain degree there exists a crystal-like order. 

In particular the growth function for spherical symmetry in three dimensions is... 

Visualization of space group symmetry in three dimensions... 

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