One learns many valuable lessons from repeated readings of E. A. Abbott's masterpiece Flatland. Probably the most lingering of these lessons is an appreciation of how very difficult it is to achieve an intuitive understanding of higher-dimensional forms from their two-dimensional shadows. We are certainly capable of achieving academic understanding of the mathematical properties of n-dimensional objects, for we prove theorems about them. Yet no matter how long we may stare even at two-dimensional representations of three-dimensional objects, we often find ourselves surprised when we examine a three-dimensional model of the depicted object.
This point is made very dramatically on the cover of Douglas Hofstadter's Gödel, Escher and Bach. Here, a pair of attractive, smoothly-carved wooden blocks is suspended and illuminated by three light sources to cast shadows on two walls and a floor. Each block casts a block E or G on one wall, a G or E on the other, and a B on the floor. Yes, that specification is precise, and one can thereby derive the ultimate appearance of blocks that would have these properties — but try imagining them in the first place!
Try also to imagine the MAA's icosahedral logo, depicted at the top of this web page, as a work of art. When MAA Executive Director Tina Straley decided to form the new Icosahedron Society, she had the vision to conceive a marvelous work of art to serve as its symbol. She commissioned the creation of solid crystal icosahedra for its elected members. Looking at an angle through any of the twenty faces (each a 3-inch equilateral triangle), one sees a display of internal reflections of the polyhedron's structure that changes dramatically as one changes one's vantage point. Whenever sunlight strikes the icosahedron, a dramatic display of pure colors illuminates many individual faces and their internal reflections even as magically sparkling images and prismatic spectra are projected throughout the walls and ceilings of the room. The Earth's rotation adds remarkable dynamism to the displays, producing a slightly different light-show every day.
As readers of his articles in Science News and his MathTrek columns know, Ivars Peterson is fascinated by mathematically-inspired art and frequently reports on it and the people who create it. In his new book, Peterson has produced ten chapters that focus on aspects of "...creativity and imagination at the intersection of mathematics and art." Fragments of Infinity tells the stories of several mathematician-artists, and produces word-picture illuminations of their architecture, tilings, sculptures, animations, paintings, drawings, weavings and structures. The writing style is immediate and personal. Peterson is personally acquainted with many of the artists he has chosen for this book and his chapters have come from often protracted dialogues with these artists. As a result, the reader is generously provided with authoritative insights into the intersection of their artistic and mathematical intentions. This book's scope fits well with its title. The author's choice of 23 color plates greatly amplifies this text.
His first chapter sets the tone of the book and serves as an overture, introducing themes that are elaborated in the chapters that follow. The following chapters discuss sculptures that reproduce theorems in stone, exploration of artistic representations extending beyond the third dimension, paper folding, fractal and mosaics, crystals, periodic and aperiodic tilings, Möbius strips, Klein bottles and braided surfaces, minimal surface sculptures, topological sculptures, and the fanciful art of M. C. Escher. The book is copiously illustrated with photos and computer images. Along the way, the reader is introduced to Costa surfaces, quasi-crystals, catenoids, pursuit curves, area-preserving transformations, horned spheres, and much more. The discussions are quite accessible, requiring no more than high school mathematics.
Peterson tells us about the many ways in which artists are using computer-based tools to increase their creativity and productivity in almost every area he covers. We learn that Heleman Ferguson uses a computer to help guide his tools in computing the amount of stone to remove, while snow sculptors in the teams Stan Wagon and Helaman Ferguson organize are able to use Mathematica to help design models for the graceful forms they enter in Colorado competitions each January. We are told to visit Thomas Banchoff's web site to see how computers can be used to represent higher dimensional objects as well as to provide a means of exploring them from different vantage points. Indeed, Fragments of Infinity identifies a vast number of ways in which modern computers can be used to experiment, explore and thus amplify artistic creativity.
There are some drawbacks to the book, however. Perhaps its most serious flaw is the publisher's choice of a matte paper that often obscures the details in Peterson's black and white photos and makes them appear muddy. This is particularly painful in the picture and detail of Ferguson's Double Torus Stonehenge: Continuous Linking and Unlinking.
Peterson tells us that much of the book's material is expanded from columns he wrote for Science News. This is most noticeable, perhaps, in the occasional repetition of information presented in prior chapters. This does allow the chapters to stand alone, to be perused at the reader's whim. The book's expectations of the reader's sophistication are also uneven: we find careful explanations of the definition and properties of p and how to work with numbers in binary — but in other sections the reader is expected to understand the properties of cross-caps, topological genus and complex exponentiation. Sometimes a definition first appears several chapters after the concept is first used.
This book begs comparison with other books on art and mathematics. Peterson's prose carries his breezy reportorial style and has generally been edited to present a seamless picture. Peterson's style is to present concepts and facts quickly and, while each chapter is written to motivate the reader, there is a minimum of technical or descriptive detail. You will not find the depth of treatment found in Keith Devlin's Mathematics: the Science of Patterns or in Banchoff's Beyond the Third Dimension. Geometric works of art are not discussed in the detail that is found in Paulus Gerdes' Geometry From Africa or explored with the spirit of fun, pun and wonderment that we remember in Doug Hofstadter's Gödel, Escher and Bach. There are no puzzles or suggestions for Gedankenexperiments such as one finds in Martin Gardner's Penrose Tiles to Trapdoor Ciphers or Knotted Doughnuts and other Mathematical Entertainments.
But there is a great deal to be gained from a book that covers so much ground so tantalizingly! By no means is this a mere coffee-table book — Fragments of Infinity is more like a tray of artistic mathematical appetizers. One is constantly lured to the back of the book to use the list of suggested readings for each chapter. Since their continuing availability is always in question, the treatment of Internet references is understandably spotty and some of the omissions are painful. For example, Peterson writes about Thomas Banchoff's hypercube film and the web site where one can tour a hypercube or a Klein bottle, but the URL is not identified in the book — if you've not seen it before, please treat yourself to a visit to http://www.math.brown.edu/faculty/banchoff.html.
While preparing to write this review, I played around with a number of online search engines, looking for relevant web sites. Stan Wagon's site, http://www.stanwagon.com, points to his photos of his teams' snow sculpture work. You can (and should!) also visit Cliff Stoll's Klein bottle business at http://www.kleinbottle.com. Surprises abound everywhere, as for example at web sites like http://www.anwg.org/resources/articles/algebra.html, where one can find Lana Schneider's discussion Algebraic Expressions as Design Elements in the weaving of Ada K. Dietz. What makes this interesting, of course, is that the host web site is that of the Association of Northwest Weavers' Guilds.
On balance Fragments of Infinity is a wonderful source of inspiration for the reader. Each chapter piques the imagination, and at least this reader was inspired to search for additional information, helped considerably by the references grouped with each of the chapters. It conveys the excitement Peterson sees in the creative spirit itself, and there is no substitute for his friends' personal insights into their art.
Marvin Schaefer (firstname.lastname@example.org) is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.