For well over a century, people have been fascinated by what it means for objects to exist in different dimensions, higher or lower than our own third dimension. This book treats a number of themes that center on the notion of dimensions, tracing the different ways in which mathematicians and others have met them in their work. Although many different branches of mathematics have used the idea of dimensions to gain new insights, it is primarily the geometers who have delighted in imagining phenomena that take place in a whole range of dimensions. Scientists, philosophers, and artists have also found inspiration by considering different dimensions, and many examples of the influence of this concept appear in the following chapters. In recent years a number of excellent books have detailed the importance of dimensional ideas in physics, philosophy, and modern art, and many of these titles are collected at the end as a set of further readings.

The book at hand represents a forty-year personal fascination with a topic that has always presented something new each time I thought I understood it. At first, dimension was only a mysterious word surprising me in a frame from a Captain Marvel comic. As Billy Batson, boy reporter, tours a futuristic laboratory, an Einstein-like figure proudly states, ``This is where our scientists are studying the seventh, eighth, and ninth dimensions.'' A thought balloon goes up from Billy Batson (and from me), ``I wonder what happened to the fourth, fifth, and sixth dimensions?'' Shortly after, a Strange Adventures comic book introduced me to the classic theme of a being from a higher dimension intruding on our world in the same way that we could intersect the flat world of the surface of a still pond. As we will see, these seemingly bizarre ideas appear again in the serious study of dimensions. Only later did I begin to appreciate the inspiration for these stories in the nineteenth-century classic Flatland, which continues to be the best introduction to the interrelationship between worlds of different dimensions.

By the time I began studying geometry in high school, I had discovered that the theme of dimensions is a thread running through all of mathematics, and into the world beyond. Architectural drawings and maps of the world try to reduce three-dimensional information to flat pages, and I became conscious both of their power and their limitations. Formulas for area and volume and formulas from elementary algebra presented patterns relating geometry in the plane to geometry in space, and again and again these patterns would invite me to consider generalizations to a fourth dimension or higher. As I learned new mathematical subjects, I always tried to see what the various notions would mean in different dimensions, but I often became frustrated at my inability to picture or model these higher-dimensional interpretations. Since the nineteenth century, people had been inventing ways to treat phenomena in higher dimensions, and I could share both their excitement about these ideas and their sense of inadequacy in imagining them.

My own opportunity to make a contribution to the visualization of higher dimensions came when I first encountered computer graphics as a young assistant professor at Brown University twenty-three years ago. The ability to see and manipulate complicated three-dimensional forms on a television screen suggested an ideal way to approach the even more complicated forms arising in higher-dimensional geometry. Much of what appears in this book is a description of how computer graphics extends our abilities to visualize different dimensions in ways that were not contemplated just a generation ago. And as this work in geometry parallels research in other fields, the insights we achieve as we investigate patterns in higher dimensions will be more and more useful to researchers in science and in art. We will see some of these influences in this volume, and even more await us in the future.

Many people have contributed to the development of this book, and I have attempted to thank them in a section of acknowledgments at the end. Particular thanks go to the students who generated the new art for this project on computers at Brown University: Davide Cervone, Nicholas Thompson, Jeff Achter, and Matthew Stone.

No book can appear without cooperation of an author and a publisher, but the staff of the Scientific American Library has been exceptional in the extent of collaboration offered throughout the project. If the writing is clear, that is due in large measure to the extraordinarily thorough efforts of editor Susan Moran, and also to the careful reading of project editor Rita Gold. Diane Maass ably supervised the final stages of proof. Credit belongs to Alice Fernandesrown for the design, to John Hatzakis for page layout, and to Travis Amos for photo selection. Susan Stetzer coordinated the production of the book. The idea for the project goes back to Neil Patterson, and the greatest credit for helping bring this book into existence goes to my editor and friend, Jeremiah J. Lyons.

Thomas F. Banchoff
April 1990

This edition features new computer graphics illustrations produced by Davide Cervone and the author at the Geometry Center, University of Minnesota. There are also several corrections in response to suggestions of helpful reviewers, in particular, H. S. M. Coxeter. For making this new edition possible, thanks go to editor Jonathan Cobb.

Thomas F. Banchoff
October 1995

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