Beyond the Third Dimension: Geometry, Computer Graphics and Higher Dimensions by Thomas F. Banchoff (review)

CURRENT MEDIA ReviewPanel:Fred Almgren, Giorgio Careri, Michele Emmer, Dawn Friedman, CraigW Johnson, Robert S. Lansdon, Clifford A. Pickover, DorisJ Schattschneider I. Book Reviews BEYOND THE THIRD DIMENSION: GEOMETRY, COMPUTER GRAPHICS AND HIGHER DIMENSIONS by Thomas F. Banchoff, Scientific American Library, New York, NY, U.S.A., 1990. 210 pp., ilIus. ISBN: 0-7167-5025-2. Reviewed lJy Michele Emmer, Via Santa Maria della Speranza 5, 00139 Rome, Italy. A first attempt to investigate fourdimensional objects by computer graphics wasmade in 1967 by Michael Noll, who described it in a short paper entitled "Displaying n-Dimensional Hyperobjects by Computers" [1]. His paper was a review ofthe mathematics for two types of projections of ndimensional hyperobjects and for n-dimensional rotations. He wrote: "Any n-dimensional hyperobject could be manipulated mathematically by a digital computer. The final threedimensional projection of the rotating hyperobject could by drawn automatically on a computer-eontrolled visual-display device as a stereoscopic movie. At first it was thought that the computer-generated movie of a fourdimensional hyperobject might result in some feelingor insight for the visualization of a fourth spatial dimension .... Unfortunately, this did not happen, and we are still as puzzled as the inhabitants ofFlatland [2] in attempting to visualize a higher spatial dimension." What was not technically possible for Michael Noll became possible for T. Banchoff and C. Strauss at Brown University at the end of the 1970s. They had the idea of using computergraphics animation to investigate the geometrical and topological properties of three-dimensional surfaces. They further wanted to look for fourdimensional objects moving in three space. In a paper written in those years, they wrote: "Real-time interactive computer graphics provides the opportunity for a research mathematician to investigate directly the geometric properties of curves and surfaces as they undergo transformations in three- and four-dimensional space." Banchoff and Strauss produced a computer-animated 16mm movie entitled "The Hypercube: Projections and Slicing". Even if all ideas used to investigate three dimensions can be generalized to arbitrary dimensions, "for four dimensions, however, it is still possible to gain a considerable amount ofgeometric intuition visually ,by interpreting projections of the vertices and edges of the four-cube in three-dimensional space." This approach to the use of computers in mathematical research was new. It became possible to construct a surface on the video terminal and move it and transform it in order to better understand its properties. In 1987 a group of mathematicians, again at Brown University, including Thomas Banchoff, realized a new computer-animated movie showing the hypersphere. Two of them wrote: 'The great potential of computer graphics as a new exploratory medium was recognized by mathematicians soon after the relevant technology became available. As display ievices and programming methods grew more sophisticated so did the depth and scope of applications of computer graphics to mathematical problems"[3] . The story is described in detail in Thomas Banchoffs book. The author, starting from various definitions of dimensions and using analogies such as the 'Square' in Flatland, introduces four-dimensional objects, in particular, the hypercube. Slicing the hypercube is one of the main techniques used in the computer-animated movie and is described in the book. Banchoffhas written: "In our days we finally have a chance to investigate the objects that our ancestors could only dream about and represent in lifeless models. Interactive computer graphics puts us into direct visual contact with slices of four-dimensional cubes. It is up to us to learn how to interpret these images, and to try to overcome the limitation of our own three-dimensional perspective." With the same technique it is possible to analyze the shadows of a fourdimensional building, as in the dream of the Square. In the chapter "Regular Polytopes and Fold-Outs", Banchoff talks of his contact with Salvador Dali. The volume is a self-sufficient essay on the various presentations offourdimensional Euclidean geometry, including a last chapter on non-Euclidean geometry. Many color pictures, including, of course, computergenerated images, are reproduced in the volume. The problem is that "it is up to us to learn how to interpret these images, and to try to overcome the limitation of our own threedimensional perspective." A...