A Popular Science book on Symmetry

This is a review of the book This Amazingly Symmetrical World by L. Tarasov. Published in 1982, it is nonetheless an all-time Russian science popularization gem issued from Mir Publishers in the last century. It is very well-written and gorgeously illustrated, with arresting pictures on almost every page. The discussions are accessible, yet quite profound and revealing.

Symmetry plays an incredibly fundamental role in physics. Nature seems to use symmetry as a way of imposing order and harmony on otherwise random phenomena. The branch of mathematics which quantitatively deals with symmetry is called group theory. This is a quite abstract, though powerful, branch of knowledge. So it is very nice to see the basic ideas spelt out for a popular audience (the book demands no more mathematical knowledge than freshman algebra and geometry).

The book is divided into two parts.

1. Symmetry in the physical universe: The first part describes the symmetries we can observe visually in nature. The book takes off with the accessible notion of mirror symmetry and the accompanying idea of enantiomorphs (mirror images which are not superimposable, like our hands).

   
   

   Translational and rotational symmetry and their combination are then considered in two dimensions. There is a gorgeous chapter on repeating two-dimensional symmetric patterns, with examples from Egyptian art and Escher.

   
   

   Moving to three dimensions brings us to a revealing discussion of the five platonic solids (cube, tetrahedron, octahedron, icosahedron, dodecahedron), including Kepler's unsuccessful attempt to model the planets using these regular (faces are the same size) polyhedra, as well as speculations about alien dice (they can't have any other shape if the probability of each face showing up has to be the same). More three dimensional examples are made of crystal gems and snowflakes, plants (apparently most flowers have five-fold rotation symmetry) and animals (which, including us, have bilateral symmetry).

   
   

   An entire chapter is devoted to the role of symmetry in assemblies of atoms and molecules, with examples such as water, methane and benzene. We also learn about the phenomenon of polymorphism, which allows the same atoms to assume different symmetries, resulting in entirely different materials (e.g. graphite and diamond). This part of the book concludes with a fascinating discussion of spirals, steroisomerism and DNA (which is a right helix).

   
   
   
   

2. Symmetry of physical laws: This is a more technical part of the book, one which refers to considerations of symmetry in phrasing the mathematical laws of nature. The idea is introduced using special relativity, which follows from the symmetry principle that the laws of physics are the same in any frame which is not accelerating.

   
   

   Further, it is discussed how translation in space or time, and rotation do not change the laws of physics (and how this leads to conservation of linear momentum, energy and angular momentum, respectively). However, mirror reflection sometimes does change the laws of physics.

   
   

   More abstract physical concepts, such as charge, spin and 'anti-particle-ness' are shown to be conserved due to more abstract symmetries. The last chapter in the book discusses the effects of symmetry in particle physics in some detail. This material could be specialized for many readers (it was for me).

In the epilog, the author points out how symmetry considerations are useful in making predictions in physics (Mendeleev predicted elements that were not yet known, Maxwell predicted the displacement current, Gell-Mann predicted fundamental particles). He also points out how the method of analogy is based on the principles of symmetry.

Afterword:

The book explains how symmetry underlies all of (especially physical and biological) existence. The book suggests also how to think of asymmetry, and how it gives character to individual structures and organisms.

One of the things I enjoyed: It assembles in one text (but distributed throughout the chapters) many famous and relevant quotes on symmetry by physicists (Einstein, Dirac, Wigner, Feynman), writers (Lewis Carroll), poets (Blake, Paul Valery), mathematicians (Weyl and Hardy), popular science writers (Martin Gardener) and architects (Le Corbusier).

An omission I found strange: there is no mention of Noether's theorem, which is a fundamental theorem linking continuous symmetries to conservation laws.