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*.

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