Mathematics possesses not only cold truth but supreme beauty, a beauty cold and austere, like that of sculpture, sublimely pure, and capable of a stern perfection, such as only the greatest art can show.
In this chapter, we will discuss the concept of symmetry in great detail. We will begin with the description of symmetry operations as coordinate transformations, followed by a discussion of the difference between passive and active operators. Then we introduce rotations, and we determine which rotations are compatible with the 14 Bravais lattices. After a discussion of operators of the first (rotation, translation) and second (mirror, inversion) kinds, we will generate combinations of symmetry operators, which will lead to glide planes and screw axes. Along the way, we will also introduce the time-reversal operator and study how it can be combined with the regular symmetry operators. We conclude the chapter with the definition of point symmetry.
Symmetry of an arbitrary object
Many objects encountered in nature none some form of symmetry, in many cases only an approximate symmetry; e.g. the human body shows an approximate mirror symmetry between the left and right halves, many flowers have five- or seven-fold rotational symmetry, …In the following paragraphs, we will discuss the classical theory of symmetry, which is the theory of symmetry transformations of space into itself.
If an object can be (1) rotated, (2) reflected, or (3) displaced, without changing the distances between its material points and so that it comes into self-coincidence, then that object is symmetric. A transformation of the type (1), (2), or (3), or a combination thereof, that preserve distances and bring the object into coincidence is called a symmetry operation. It should be clear that translations can only be symmetry operations for infinite objects. The word “symmetric” stems from the Greek word for “commensurate.” Note that the identity operator (i.e., not doing anything) is also considered to be a symmetry property; therefore, each object has at least one symmetry property.