I would like to thank a number of people who have provided me with inspiration, assistance, and support over the years and who have contributed directly or indirectly to the writing of this book.

My earliest mathematical encouragement came from my mother, a kindergarten teacher, and my father, a payroll accountant. Herb Lavine, three years older than I, taught me algebraic patterns on the tops of cartons in his father's grocery store. In my freshman year at Trenton Catholic Boys' High School, Father Ronald Schultz listened to my first theorem--calculating when advancing shadows would bisect the triangular tiles on the church floor--and he also paid attention to my first theory relating the fourth dimension and the Trinity. William Hausdoerffer at Trenton State gave the first college lecture I ever heard, and today he remains my consultant on the subject of sundials.

At the University of Notre Dame my advisors, and eventually my colleagues and friends, were Frank O'Malley in writing, R. Catesby Taliaferro in mathematics, Dean Charles Sheedy, Arnold Ross, who gave me my first chance to teach at his summer program for high school teachers, G. Y. Rainich, who introduced me to non-Euclidean geometry and relativity, and Father Robert Pelton and Father John Dunne, who gave me the opportunity to explore geometry and theology.

My Ph.D. advisor at the University of California, Berkeley, Professor Shiing-Shen Chern, and my primary geometric advisor, Nicolaas Kuiper, encouraged me while I studied the geometry of polyhedral surfaces in higher dimensions. As a Benjamin Peirce Instructor at Harvard I had my first chance to teach the fourth dimension. I appreciate my contact there with William Reimann and William Wainwright at the Carpenter Center for the Visual Arts, and with my friends Edwin Moise and George Hunston Williams.

At Brown University I thank particularly Charles Strauss, my collaborator in computer graphics and geometry for more than twelve years, and Harold Weber, who built the first device that let us see four-dimensional objects in real time. Many Brown University colleagues have contributed directly to the development of the ideas in this book, including Fred Bisshopp and Philip Davis (applied mathematics), Hunter Dupree (history), Richard Fishman (art), Richard Gould (anthropology), Peter Heywood (biology), John Hughes (mathematics), Hüseyin Koçak (applied mathematics), Martha Mitchell (archives), Doctor Alfred Moon (radiology), Henry Pohlmann (mathematics), Joan Richards (history), Karen Romer (dean), James Schevill (English), Gerald Shapiro (music), Merton Stoltz (provost), Julie Strandberg (dance), James Van Cleve (philosophy), Andries van Dam (computer science), Tom Webb (geology), and Arnold Weinstein (comparative literature). I am grateful for the constant help of the mathematics office staff--Dale Cavanaugh, Carol Oliveira, and Natalie Johnson (and her father). Thanks also to my colleagues at different institutions--Antony Raubitschek (classics), Joan and Erik Erikson (psychology), and John Tukey and Paul Tukey (exploratory data analysis), and the members of the Clavius Group. I would like to acknowledge the National Science Foundation, the Office of Naval Research, and the Mathematical Sciences Education Board for their support of several projects described in the book. My geometric collaborators Peter Giblin, Wolfgang Kühnel, Ockle Johnson, and Clint McCrory were patient while I worked on this book. I remember especially my late geometry colleagues Hassler Whitney, William Pohl, and Stephanie Troyer.

All artists involved with dimensions in art acknowledge the contributions of David Brisson at the Rhode Island School of Design and founder of the Hypergraphics Group. I thank him and the other artists whose work appears in this book: James Billmyer, Salvador Dalí, Attitio Pierelli, Lana Posner, Tony Robbin, and Jose Yturralde. I would also like to thank artists Nieves Billmyer, Harriet Brisson, Arthur Loeb, Colin Low, Michele Emmer, Charles Eames, Malcolm Grear, C. C. Beck (who drew the Captain Marvel comic that started it all), and all my friends at the Providence Art Club, especially David Aldrich, William Gardner, Carlton Goff, Garvin Morris, Maxwell Mays, Raymond Parker, and Thomas Sgouros. I acknowledge the influence of many conversations with my fellow writers about higher dimensions, including Dionys Burger, H. S. M. Coxeter, Alexander Dewdney, Henry Thomas Dolan, Martin Gardner, Linda Dalrymple Henderson, Madeleine L'Engle, and Jeff Weeks. Special help for this project came from my friends in England--Sir Basil Blackwell, William Hallett and Terry Heard at the City of London School, Barbara Phillipson, and David and Deborah Singmaster. Many friends have listened to me talk about this book over the years, in particular Donald Albers, Doctor Frederick Barnes, Carl Bridenbaugh, Daniel Driscoll, James Fitzwater, Ambrose Kelly, Margaret Langdon Kelly, Judge William Mackenzie, Louise Mackenzie, David Masunaga, Thomas Roberts, and Allen Russell, and during this past year, Peter Chase, Harold Ellsworth, Bishop John Higgins, and Robert Morehead Perry.

In the illustration acknowledgments I will individually credit the efforts of the many students who have contributed to our work in geometry and computer graphics at Brown University. Finally I would like to thank all the students who have participated in my courses on the fourth dimension over the past twenty-five years, and especially those who have been my assistants for those courses, including Michael Holleran, Steven McInnis, Lindley Gifford, David Pinchbeck, Brandt Goldstein, David Goldsmith, Anne Morgan, Michael Chorost, Ilise Lombardo, Eric Chaikin, and now David Burrowes and Matthew Salbenblatt. They will recognize many of their own ideas in this book.

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