Further Readings

An excellent reference for the relationship of physics and dimensions is the recent Scientific American Library volume, A Journey into Gravity and Spacetime by John Archibald Wheeler (W. H. Freeman, 1990). Two books by Rudolf Rucker also present the relationship between geometry and relativity: The Fourth Dimension: Toward a Geometry of Higher Reality (Houghton-Mifflin, 1984), and Geometry, Relativity, and the Fourth Dimension (Dover, 1977). The former book also treats the philosophical and mystical side of the subject, not only through Rucker's own ideas, but by incorporating the writings of Charles Howard Hinton, whose work Rucker collected in Speculations on the Fourth Dimension: Selected Writings of C. H. Hinton (Dover, 1980). Jeff Weeks has written a fine account of current work in the topology of three-dimensional spaces in The Shape of Space (Marcel Dekker, 1985).

The ideal resource for the history of geometry in the nineteenth century is the recent book by Joan Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England (Academic Press, 1988), which contains an excellent bibliography. A historical survey of higher dimensions appears in the introduction to Geometry of Four Dimensions, written by Henry Parker Manning in 1914 and reprinted by Dover in 1956. See also the preface of Manning's book The Fourth Dimension Simply Explained (Munn and Company, 1910; reprinted by Dover, 1960).

The definitive book on higher dimensions and art is Linda Henderson's volume The Fourth Dimension and Non-Euclidean Geometry in Modern Art (Princeton University Press, 1983). Another good resource on the subject is Hypergraphics: Visualizing Complex Relationships in Art, Science, and Technology, edited by David Brisson and containing the pioneering paper of A. Michael Noll as well as many other articles (Westview Press. 1978).

Martin Gardner has featured the fourth dimension in many of his Scientific American columns, and these have been anthologized in several volumes of his collected works, including The Unexpected Hanging (Simon and Schuster, 1986), Mathematical Carnival (Alfred A. Knopf, 1975), and The New Ambidextrous Universe (W. H. Freeman, 1990). Alexander Dewdney has also written articles on dimensions in his Scientific American columns. Particularly interesting is his treatment of two-dimensional science and technology, which is discussed at length in his allegory The Planiverse (Poseidon Press, 1984). Dewdney's column ``Computer Recreations'' in the April, 1986, issue of Scientific American featured the animation of the hypercube.

Ivars Peterson has included in his book The Mathematical Tourist (W. H. Freeman, 1988) a report of the conference on hypergraphics at Brown University on the occasion of the centennial of Flatland in 1984. There have been several new editions of Flatland during the past ten years, and the best-selling Dover edition, first printed in 1952, has a good introduction by Banesh Hoffman. A new edition by Princeton University Press appeared in Spring, 1991, with an introduction written by Thomas Banchoff. A sequel to Flatland called Sphereland by Dionys Burger, was first published in 1965 and was reprinted by Harper and Row in 1983.

For fractal geometry there are several very good sources, including The Fractal Geometry of Nature by Benoit Mandelbrot (W. H. Freeman, 1982), The Beauty of Fractals by Heinz-Otto Peitgen and P. H. Richter (Springer-Verlag, 1986), and Fractals Everywhere by James Barnsley (Academic Press, 1988).

General references on the topology of surfaces include the classic Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen (Chelsea, 1952) and A Topological Picturebook by George Francis (Springer-Verlag, 1988).

The specific reference for the dynamical systems related to pendulum motion is the paper ``Topology and Mechanics'' by Hüseyin Koçak, Fred Bisshopp, David Laidlaw, and the author in Advances in Applied Mathematics, Vol. 7, pages 282-308, 1986. A good reference for singularity theory in geometry is Curves and Singularities by Peter Giblin and James Bruce (Cambridge University Press, 1984).

For polyhedra the standard references are the books of H. S. M. Coxeter: Regular Polytopes (Dover, 1973), Regular Complex Polytopes (Cambridge University Press, 1974), and Introduction to Geometry (John Wiley and Sons, 1961). The classic book by D. M. Y. Sommerville, Geometry of n Dimensions (Methuen, London, 1929; reprinted by Dover, 1958) is also an excellent reference. Another, more formal source is Convex Polytopes by Branko Grunbaum (Interscience Publishers, 1967). More recent books are Arthur Loeb's Space Structures: Their Harmony and Counterpoint (Addison-Wesley, 1976) and Shaping Space, edited by Marjorie Senechal and George Fleck (Birkhäuser, 1988). Another reference for the coordinate geometry of four dimensions is Linear Algebra through Geometry by Thomas Banchoff and John Wermer (Springer-Verlag, 1983, second edition, 1991).

The premier works on the visualization of data are Edward Tufte's The Visual Display of Quantitative Information (Graphics Press, 1983) and John Tukey's Exploratory Data Analysis (Addison-Wesley, 1977). Also by John Tukey and cowritten by Paul Tukey is a fine article ``Graphic Display of Data Sets in III or More Dimensions,'' which appeared in Interpreting Multivariate Data (John Wiley and Sons, 1981).

Several works of fiction are extremely insightful, including Madeleine L'Engle's A Wrinkle in Time (Farrar, Straus, and Giroux, 1962), which introduced generations of readers to the concept of the tesseract. Robert Heinlein's story ". . . and He Built a Crooked House" appears in Clifton Fadiman's collection Fantasia Mathematica (Simon and Schuster, 1958), along with other mathematical stories.

Finally, The Hypercube: Projections and Slicing is available on film and videotape from International Film Bureau, 332 South Michigan Avenue, Chicago, Illinois 60604. Videotapes of The Hypersphere: Foliation and Projections and Fronts and Centers are distributed by The Great Media Company, P. 0. Box 598, Nicasio, CA 94946. For ongoing information about the author's visualization projects, contacthttp://www.geom.umn.edu/~banchoff on the Internet. Professor Banchoff's home page contains many examples of interactive computer graphics illustrations related to the material in Beyond the Third Dimension.

[Up] Table of Contents