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Symmetry Discovered: Concepts and Applications in Nature and Science

Joe Rosen
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
David A. Huckaby
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This book is an excellent introduction to symmetry designed for general readers, but especially appropriate for advanced high-school students and lower-level undergraduates. In the preface, the author gushes about his love for symmetry and expresses his hope that the reader will “catch the symmetry disease.”

The entire book — until the final chapter — is devoted to introducing various types of symmetry. Geometric symmetry receives the most thorough treatment, but since one of the author’s goals is to convince the reader of symmetry’s pervasiveness, he also discusses various other types of symmetry, such as temporal symmetry, permutation symmetry, and analogy. Many real-life systems have only approximate symmetry; so when the author introduces a new symmetry, he typically also discusses the approximate version that a practitioner is likely to encounter.

As befits a book on this topic, diagrams abound. The writing is crisp. The long chapter on geometric symmetry, in which the various symmetries are given precise verbal descriptions along with accompanying diagrams, provides the novice student an excellent opportunity to practice careful reading. Indeed, students would do well to read these descriptions of geometric symmetries, draw their own diagrams to match the descriptions, and then compare their diagrams with those provided in the book.

Students should also take advantage of the opportunity to learn a bit of advanced mathematics in the form of group theory. Basic group theory is introduced early on; throughout the remainder of the book, each topic is first treated informally and then framed in group-theoretic terms. All paragraphs involving group theory are clearly marked, and readers are invited to skip these paragraphs if they wish. But since elementary geometry is such an accessible medium through which to be exposed to group theory, intrepid readers are encouraged not to forego this opportunity to learn a little abstract mathematics.

Having spent most of the book discussing various types of symmetry, the author considers in the final chapter how symmetry can be used to study cause-and-effect relationships in nature. The principal tool here is the symmetry principle. (Stated roughly: An effect has at least as much symmetry as its cause.) Armed with this principle, we can solve problems of the following form: Given a certain cause in nature, what is its effect? Several such problems are posed and then solved using the symmetry principle. (For example, given Newton’s first law, symmetry is used to show that a planet’s orbit around the sun lies in a plane that goes through the center of the sun.) The last section of the chapter briefly discusses the use of symmetry in addressing the reverse problem, which often obtains in basic research: Given a certain observed effect, what is its cause? This section includes a brief discussion of the roles of beauty and simplicity in scientific theories.

This book is suited for self-study. Students who work through it on their own should try to solve the many problems posed throughout the text. The Dover reprint edition includes an appendix with answers to many of these problems. Also scattered throughout the text are copious references to the literature, providing pointers for further reading to the student who wants to delve deeper into a certain topic. The bibliography is extensive for a book at this level; the Dover reprint edition includes an addendum that lists a few more recent references.

The original edition of this book has been enjoyed by many. If the love of symmetry is indeed a disease, the Dover edition, with its extra features, should help spread the contagion even further. The precise and lively writing, along with the many diagrams, help make this book an excellent introduction to symmetry.

David A. Huckaby is a professor of mathematics at Angelo State University.



  Preface to the Dover edition
  1 Symmetry: what? where? how?
    What is symmetry?
    Where is symmetry?
    How is symmetry?
  2 The language of symmetry
    Concepts and terminology
    Group theory (optional)
  3 Geometric symmetry
    Linear symmetry
    Planar symmetry
    Spatial symmetry
  4 Other symmetries and approximate symmetry
    Temporal symmetry
    Permutation symmetry
    Color symmetry
    Approximate symmetry
  5 Symmetry in nature
    Symmetries with which we are acquainted
    Symmetries we have no yet studied
  6 Uses of symmetry in science
    The symmetry principle
    The symmetry approach
    Symmetry in research
  Addendum to Bibliography
  Solutions to problems