*Symmetry*is a slender work which focuses on the mathematics of symmetry, starting simply but building up to increasing abstraction, but which also explores more widely, touching on aspects of symmetry in philosophy, art and architecture, physics, and biology. Weyl was a leading mathematician, physicist and philosopher of the first half of the twentieth century and he offers a magisterial perspective.

*Symmetry* begins with bilateral symmetry, looking at its uses in art
and architecture and at left-right symmetry in biology, in phylogeny
and ontogeny. The only mathematics introduced in the first lecture,
and that only in passing, is the notion of congruence.

The second lecture moves on to translational and rotational symmetries, looking at frieze and cyclic patterns in art and architecture, and in the natural world at flowers, echinoderms, snowflakes, and so forth. The concepts of automorphisms and groups are introduced here and Weyl presents a classification of the finite groups of proper and improper rotations in space (though proof of the completeness of this is left to an appendix).

Lecture three is devoted to "Ornamental Symmetry", or the symmetries of regular tilings of the plane. This introduces some basic linear algebra and the notion of a lattice, and works through classification of the 17 plane symmetry groups. It doesn't attempt "an explicitly algebraic description" of those groups, but it is still fairly involved.

It does include a discussion of symmetry in physics, but the final lecture on "The General Mathematical Idea of Symmetry" mostly offers further mathematics. Almost as if to scare off most readers, it opens with a paragraph on the classification of "the unimodularly inequivalent discontinuous groups of non-homogeneous linear transformations which contain the translations with integral coordinates but no other translations". And it goes on to touch on Galois theory.

The limitations of *Symmetry* are fairly obvious. Its illustrations,
while they adequately illustrate the connections being made to art,
architecture and biology, are grainy black and white halftones which
hardly grab the attention. And Weyl is writing for, or speaking to,
a relatively sophisticated audience. He introduces mathematical ideas
rapidly, more as a refresher than a genuine introduction, and his
presentation sometimes seems a little awkward; some of his terminology
is also a little dated. So this is really not recommended as a first
introduction to group theory.

General readers might enjoy the first half of *Symmetry*, however,
and wing their way through the second half, while anyone with some
familiarity with group theory, and curious about a broad perspective
on its application to symmetry, should find the whole work engaging.

March 2016

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