Computers, Pattern, Chaos and Beauty: Graphics From an Unseen World by Clifford A. Pickover (review)

tortions were tempered by what our architects know as axonometric perspective . The procedure goes back to Johann Heinrich Lambert's 'military or soldier's perspective' of the seventeenth century, the parallel projection described in later treatises as isometrical. On the other hand, curvilinear perspective, which has recently aroused new attention, was used in practice by artists of the early Renaissance such as Konrad Witz and Johan Fouquet, and is traced back by Kemp in the theoretical literature to a German treatise of 1624 (p, 247). From the beginning, the representation of depth by geometrical construction was rivaled by the purely mechanical projections best known from the devices illustrated by Diirer, a tradition continued through the uses of the camera obscura, the camera lucida, and, finally, photography. I notice that Kemp still attributes the invention of geometrically constructed perspective to Brunelleschi even though we know for certain only that he traced architectural projections by the entirely ungeometrical use of a mirror. Kemp has useful sections on the role of illumination in the representation of spatial volume and on quadratura, that is, the reduction of the human form to stereometric elements . Galileo ascertained the mountainous relief of the moon by the varying shadows he observed through his telescope, and in painting the artists concerned with stereometric shapes emphasized their structures byappropriately cast shadows. In a geometrically inclined artist like Poussin, this rigidity of structure was counteracted by the chiaroscuro effects, which ever since Leonardo modified form by a painterly softening. The quadratura, reemerging in the cubist shapes of modern artists such as Fernand Leger, serves to analyze the complexity of the human body in the treatises of Diirer and later of Lamazzo. Rubens, who attempted in a drawing to reduce the Farnese Hercules to a system of cubes, noted in his pocket book that the cube suits the construction of the "massively robust male figure , while the circle pertains more properly to female anatomy" (p. 99). Kemp's very ample treatment of color theory cannot but suffer from the inability of small color plates to illustrate the nuances of colorist painting . We read that the flesh shadows on Heliodorus's leg in Delacroix's painting "are laced with green, adorned with fiery tongues of pure red and surrounded by contours of a violet hue" (p. 310), but the illustration lets us down. Theoretically, the outstanding theme is the Aristotelian tradition according to which colors come about by "the mixing in different ratios of the contrasting properties of whiteness and blackness" (p. 264), a belief that still lingers in Goethe's doctrine of color. So many technical and historical details are offered in this comprehensive book that readers who are mostly after generalities are tempted to skip over them. This practice is discouraged by Kemp's curious habit of using personal pronouns even though the names of the persons to whom they refer have been mentioned as far away as a paragraph or so earlier. But a conscientious adherence by the reader to Kemp's context does bring reward. COMPUTERS, PATTERN, CHAOS AND BEAUTY: GRAPHICS FROM AN UNSEEN WORLD by Clifford A. Pickover. St. Martin's Press, New York, NY, U.S.A., 1990. 391 pp., illus. Trade, $29.91. ISBN: 0-312-04123-3. (Simultaneously published by Alan Sutton Publishers, Glouchestershire, U.K., 1990. ISBN: 0-86299-792-5.) Reviewed by RoW F. Malina, Center for EU,/ Astrophysics, University of California, Berkeley, CA 94720, U.S.A. This book is an exploration and documentation of the visual presentation of calculations of mathematical equations using computer graphics systems . Readers of Leonardo will be familiar with the striking images from fractal mathematics and chaos theory. This book provides recipes not only for these areas of visual mathematics but also from complex variables, number theory, Markov processes and other mathematical formulations. Readers are provided with sufficient technical detail to be able to program their own home computers to generate the images displayed in the book. Imagery is the heart of this book, and Clifford Pickover is like a naturalist exploring a new visual world, until recently the province only of the mathematician. He collects interesting specimens. He uses his 'butterfly net' to reprocess the image...