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Hans Walser
Mathematical Association of America
Publication Date: 
Number of Pages: 
[Reviewed by
Michele Intermont
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This recent publication by the MAA, a translation of Hans Walser's 1996 book, is a brief (85 pages) and informal look at symmetry. As the author states in the forward, "The object of this book is to present selected examples of symmetry in an understandable way... For me it is most important to 'sharpen the eye' for the proper perception of symmetry in the world around us." Towards this end, Walser chooses just a few examples of symmetry to expound upon. His selection of topics and viewpoint on those topics is different from many authors on the subject.

There are six chapters in the book. The first, Little Mirror, Little Mirror, begins with an experiment — reflecting a small mirror in another mirror. The reflected image is again reflected and so forth, and the author tantalizes us by revealing that the lengths of these images leads to, not the geometric series we were perhaps expecting, but to the harmonic series. This chapter also includes a discussion of fractal images. The second chapter, Inside and Outside, continues the theme of reflections, this time reflecting in a circle. One doesn't often equate reflected images under such a transformation with symmetry, and in this the author does succeed in sharpening the eye to symmetry. This chapter ends with exhortations to the reader to think about reflections in polygons. Symmetric Procedure, Chapter Three, is concerned with another atypical example of symmetry. Here, finding the center of gravity is exemplified as a symmetric procedure. But why stop at considering the center of gravity of distinct points? The author challenges us to find a procedure which computes the "edge center of gravity" for triangles and quadrilaterals.

Ah, would a book on symmetry be complete without at least a few words on tiling? Probably not, and in Chapter Four, Parquet Floors, Lattices and Pythagoras, Walser gets as close as he will come. This is the most extensive chapter in the book, and in it the reader is asked to consider how tilings of squares can lead one to a proof of the Pythagorean Theorem. Also studied is the symmetry that sometimes arises from superimposing two square grids.

The Problem of the Center follows. Of course, one can think about rotations and reflections when one thinks about the notion of center, but here we are led to consider the center differently. In light of Chapter Three, it is fitting that the author asks us to think about the center "dynamically as the collision-point of two opposing motions." The idea of center also leads to discussions of mean values. The final chapter is entitled Symmetry in Word, Script and Number. According to the translators (Peter Hilton and Jean Pederson), it is much pared down from the original German text. This is understandable, but still disappointing. There is isn't much left - just a brief discussion of palindromes and palindromic numbers and a few words on rhyming schemes.

The style in which this book is written deserves mention. The author expects interaction between the reader and the material, and demands it by posing questions throughout. Some of these questions are straightforward, some are intriguing although not straightforward. This style is part of what will charm many about the book, and rightly so. The questions have the capacity to engage the reader, and I can easily envision some readers finding themselves absorbed in a few of them. Of course, this style will also frustrate some. For those, I mention that Walser does include answers to the questions at the end of each chapter.

This interactive style reminded me of David Farmer's book Groups and Symmetry: A Guide to Discovering Mathematics. I have in fact, used Farmer's book as a text for class, and while Walser's book is an enjoyable diversion, I will continue to look to Farmer's book in classroom situations. Partly this is my own bias, as Farmer's goal is to introduce some group theory and I am always happy to have opportunities to do that. Partly, however, I found that Walser's writing caught me so off-guard at times that I did find myself becoming frustrated.

Overall, Walser has done a fine job in creating an interesting kaleidoscope of ideas about symmetry. His view of the subject is non-standard, and for that reason many inquiring minds will enjoy it. He peppers his writing liberally, not only with (as already mentioned) questions for exploration, but also with references. In this way he manages to keep his treatment succinct without shortchanging the reader.

Michele Intermont ( is Assistant Professor of Mathematics at Kalamazoo College. Her area of specialty is algebraic topology.
  1. Little mirror, little mirror: Even further inwards; The mirror in a mirror in a mirror; An avenue of poplars; The monitor in the monitor; As seen from the side.

  2. Inside and outside: Reflecting in a circle; Composition of two-circle-reflections; Direct construction of the image point; Circle-reflection invariants; Image of a straight line; Representation in Cartesian coordinates; Image of a circle; Square-reflection; Other reflection.

  3. Symmetric procedure: Center of gravity in the triangle; Center of gravity in the quadrilateral.

  4. Parquet floors, lattices, and Pythagoras: Parquet floors; Parquets and Pythagoras; Construction of a proof-diagram; Other cathetus-figures; Overlapping of lattice-points; Pythagorean triangles; Parametrizing the primitive triangles; In a regular triangular lattice. The problem of the center: Where is the center of the world?; Mean values; Half is eaten; Average speed; Correcting systematic errors; Minimal service-routes; Symmetry in word, script and number: Palindromes; Palindromic numbers; Rhyming schemes.