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Mathematics and Art: Mathematical Visualization in Art and Education

Claude P. Bruter, editor
Publication Date: 
Number of Pages: 
Mathematics and Visualization
[Reviewed by
Annalisa Crannell
, on

This book must have been a delight to put together. It contains 28 essays by 25 authors from 8 countries, and covers at least 5 disciplines (mathematics and art, of course, but also computer science, film, and music). The authors themselves are giants in their fields. There are artists (Charles Perry, John Robinson, Patrice Jeener, Dick Termes) whose sculptures and engravings appear worldwide. There are mathematicians whose books have changed the way we look at our subjects (and I use "look" in the literal sense); or whose videos appear in almost every mathematical library.

The book is edited by Claude Bruter, one of the founders and leaders of the ARPAM project — the Association pour la Réalisation et la Gestion du Parc de Promenade et d'Activités Mathématiques, or Association for the Realization of the Mathematical Park. (If you read French, you can learn more about this at It is Bruter who kicks off the book with his "Presentation of the Colloquium". His introduction just begs to be read with a French accent:

... You will understand that he asked me the origin of this problem [of sphere eversion], since the work of surgeons would be greatly made easier if they could turn us over like gloves. There is nothing less sure than that the mathematicians [must] take care of helping the surgeons. Usually, it is rather the contrary that happens. [page 7]

The strength of this collection of works is that it comes from a collection of talented people who have a lot to say on the subject of visualization in mathematics and art.

Alas, the weakness of this collection of works is that it is not quite clear whom they are saying it to.

As is sometimes the case with a Proceedings volume, the articles vary considerably in tone, length, and audience. The shortest two articles are 2 and 4 pages, and are essentially CVs with photographs of artwork. The longest article is 29 pages. Many mathematically-based articles completely avoid formulas or other rigorous details; one of the mathematical articles carefully defines "root" and "polynomial"; another uses normed Banach spaces in a significant way; yet another launches in with "the [discrete minimal catenoid] is a critical point of the discrete area functional," and proceeds from there to theorems about four-parameter families of these catenoids.

The order of the articles is the order in which the authors spoke at the colloquium, not at all in order of topic. For example, the three (excellent) articles on math and music are the 5th, 9th, and 12th in the book.

Perhaps more confusing is the "insider-ism". There are no bibliographies of the authors, which might have been helpful to the readers who are new to the field. Many of the articles contain — or are entirely — philosophical or historical reflections on projects the author has undertaken, and of these articles, several neglect to introduce the project itself. It is easy to imagine that some of the authors are talking to one another, and not to the readers of the book. (For example, Bruter discusses the justification for the ARPAM project, but does not explain the acronym, the location of the project, who founded the project, or who is currently involved in it). For this reason, the book will be more helpful to people who have already been swimming in the waters of math and art for a while.

If the word "Education" in the subtitle encourages you (as it did me) to consider this as a textbook, you (and I) should think again. In this book, "Education" applies perhaps as much to museums as to classrooms. Still, there are some articles in here that are accessible to undergraduates at a variety of levels, and that might be useful as modules within a course or as the starting point for independent study projects.

Since I mentioned "swimming in the waters of mathematics and art", let me mention some of my favorite "sunken treasures" in this book.

Music  *  Solid Geometry  *  Symmetry  *  Sphere Eversions  *  Education

Music. The three articles on mathematics and music are lovely. Eric Neuwirth carefully describes a thorough mathematical history of tuning instruments. He includes Mathematica code that allows the programmer to "hear" the mathematics and to experiment with it. Carlos Simões describes the algebra behind Schoenberg's 12-tone construction. Yves Hellegouarch follows up on Simões' article with the algebra (and what I think of as "near-miss algebra") of expressive intonation-how singers can get away with breaking the algebraic rules behind our usual mathematical scale.

Solid Geometry. George Hart's article includes a step-by-step description of building polyhedral sculptures, both via computer construction and via physical construction. This article should be of interest to anyone interested in platonic solids and their relatives. Hart is a prolific mathematician who has written books on Zome tools and Multidimensional Analysis, so it is no surprise that this article has lots of pictures accompanied by specific explanations of what works and what problems to avoid.

Symmetry. I am a sucker for dynamical systems and fractals, so it's no surprise that Michael Field's article on symmetrical attractors of dynamical systems caught my eye. Field, an author/co-author of 8 books, pays attention to both the mathematics of his systems (how the algebra of symmetries fits in with the complex dynamics), and also to the aesthetics of a pleasing design. The designs he presents will be familiar to those who often read the Notices, or who have seen the Math Awareness Week poster from 1995. Scott Crass's article, "Solving Polynomials by Iteration" should be read before Field's, because it begins at the beginning (defining roots) and moves into more detailed mathematics. Crass may well be distinguished among this distinguished crowd of authors by his problem that appeared on NPR's "Car Talk".

Moving out of dynamical systems, Maria Dedo describes how to explore symmetry using mirrors in the classroom and in museum exhibits (see In this article, she does a nice job of describing Coxeter groups, and building from there to describing how we can use mirrors to experiment with symmetries in R2 and in R3. I particularly liked the constructions of R3 symmetries, and the way she ties the mathematical definitions to the physical mirror boxes and objects within.

Sphere Eversions. The longest and most mathematically technical article in the book is by François Apéry, a mathematician from Alsace. I would love to read this article with an advanced undergraduate. Apery describes how to visualize a sphere eversion using the "principle of least action" on wire models. To do this, he begins with describing a particular norm on a Banach space of C2 paths in R3. (I did mention that the article was mathematically technical). After developing the mathematical background and theory, which he does carefully and clearly over a dozen pages, he then uses this theory to explain the physical wire models. He includes pictures, explanations of which materials he used, and methods of construction. His description is something that is probably well known to geometers, but was new to me, and I was fascinated.

A nice companion piece to this follows directly; John Sullivan describes sphere eversions. His article contains less in the way of rigorous mathematical detail, but has many really lovely pictures accompanied by general explanations of the eversion processes.

Education. If you are thinking of developing a course or an exhibit on mathematical visualization, the experiences of Michael Field (okay, I'm trying not to sound like a groupie, but he is a very good writer) and of Ronnie Brown might interest you. Brown describes a course on knots that led to a traveling exhibition; Field explains how he used his software program PRISM to develop an interdisciplinary course for art students, and then for local middle and high school math teachers. Sandwiched in between these two papers is the description, by Manuel Arala Chaves, of a web-based exhibit called "Atractor" — see (the single "t" in "atractor" is because the exhibit is based in Portugal).

Conclusion. Although the articles in this book are uneven, the book itself is like the little girl who had a little curl right in middle of her forehead. When the articles are good, they are very, very good. A more severe editing would have improved the rest of the book significantly.

Annalisa Crannell's primary research is in topological dynamical systems, but she is also active in developing curricular materials for courses on "Mathematics and Art" as well as materials for writing across the curriculum.