Symmetry

Description

Symmetry is a classic study of symmetry in mathematics, the sciences, nature, and art from one of the twentieth century's greatest mathematicians. Hermann Weyl explores the concept of symmetry beginning with the idea that it represents a harmony of proportions, and gradually departs to examine its more abstract varieties and manifestations—as bilateral, translatory, rotational, ornamental, and crystallographic. Weyl investigates the general abstract mathematical idea underlying all these special forms, using a wealth of illustrations as support. Symmetry is a work of seminal relevance that explores the great variety of applications and importance of symmetry.

cited Cantwell's Introduction to Symmetry Analysis (p. 4, PDF p. 48):

*Definition 1.1. An object is symmetrical if one can subject it to a certain operation and it appears exactly the same after the operation. The object is then said to be invariant with respect to the given operation. *

Excellent popular introduction to Lie theory, symmetry, etc.

This is slightly modified version of the Louis Clark Vanuxem Lectures given at Princeton University in 1951, with two mathematical appendices. The first lecture begins by showing how the idea of bilateral symmetry has influenced painting and sculpture, especially in ancient times. This leads naturally to a discussion of "the philosophy of left and right'', including such questions as the following. Is the occurrence in nature of one of the two enantiomorphous forms of an optically active substance characteristic of living matter? At what stage in the development of an embryo is the plane of symmetry determined?
The second lecture contains a neat exposition of the theory of groups of transformations, with special emphasis on the group of similarities and its subgroups: the groups of congruent transformations, of motions, of translations, of rotations, and finally the symmetry group of any given figure. Groups containing translations in only one direction are illustrated by the metamerism of "potentially infinite'' plant shoots and animals with very many legs, also by the rhythm of poetry and music and by band ornaments. In the same spirit, the cyclic and dihedral groups are illustrated by snowflakes and flowers, by the animals called Medusae, and by the plans of symmetrical buildings. Similarly, the infinite cyclic group generated by a spiral similarity is illustrated by the Nautilus shell and by the arrangement of florets in a sunflower.
The third lecture gives the essential steps in the enumeration of the seventeen space-groups of two-dimensional crystallography, as first carried out by Pólya and Niggli [Z. Kristallographie 60, 278–298 (1924)]. The author remarks that examples of all these groups occur in the decorative patterns of the ancient Egyptians, as well as in those of the Arabs [cf. Edith Müller, Thesis, University of Zürich, 1944; MR0038977].
The fourth lecture begins with the complete list of finite groups of congruent transformations in Euclidean 3-space, and the attenuated list produced by the crystallographic restriction. The author mentions the 230 space-groups, and Laue's discovery that the arrangement of atoms in a crystal is revealed by X-rays. In his analysis of physical space, he distinguishes between the group of "geometric automorphisms'' (generated by reflections, rotations and dilatations) and the smaller group of "physical automorphisms'' (congruent transformations). The latter lacks the dilatations because the elementary particles determine an absolute length. He shows how the special theory of relativity is essentially the study of the inherent symmetry of the four-dimensional space-time continuum, where the symmetry operations are the Lorentz transformations; and how the symmetry operations of an atom, according to quantum mechanics, include the permutations of its peripheral electrons. Turning from physics to mathematics, he gives an extraordinarily concise epitome of Galois theory, leading up to the statement of his guiding principle: "Whenever you have to do with a structure-endowed entity, try to determine its group of automorphisms''.
In the first appendix he enumerates the finite groups of rotations in three dimensions. In the second he shows very simply how this enumeration can be extended to the finite groups of congruent transformations (including reflections and rotatory-reflections).

Reviewed by H. S. M. Coxeter