Differential and Difference Equations through Computer Experiments, and: Phaser: An Animator/Simulator for Dynamical Systems for IBM Personal Computers by Hüseyn Koçak, and: Fields & Operators by Martin Lapidus, and: Models by Martin Lapidus (review)

providing a thorough tutorial for the software. In addition to periodic 'all-over' tilings and patterns, each of the three programs discussed above can produce some other periodic designs-those with only rotation symmetry (symmetry group Cn) , only rotation and reflection symmetry (symmetry group Dn) , and periodic 'strip' or border patterns. Symmetry Studio produces Cn and Dn patterns for 1 ~ n ~ 99 and all seven types of periodic border patterns . Crystal Paint only produces a few of the Cn and D, patterns, and Escher-Sketch does not make these but produces all types of periodic border patterns. Mosedit is a program that is intended not only for design but for mathematical research. The authors are at the School ofArchitecture and Industrial Design, University of Montreal . To use the program intelligently, the user should know the rudiments of classification of tilings by isohedral type [4]-this is a much finer classification than by symmetry groups. In this program, the user selects one of 46 'fundamental' isohedral types from a menu (each illustrated as a polygon showing 'free' edges and symmetry elements); a polygonal tiling with an 'F' motif displays the symmetry of the chosen type. The user can then design a single tile using the mouse, guided by the shape of a sample polygon and its symmetry elements . When the tile is complete, the program can automatically draw the corresponding isohedral tiling. There are many other options for investigation -for example, the shape of the tile can be modified in various ways, a dual to a given tiling can be produced , tiles may be colored so as to create color symmetries, designs may be drawn inside the tiles. The user interface is necessarily more complex than that in the other programs. In all of these programs, the uninitiated user can.just by playing around and observing the generation of patterns , gain an understanding of some of the fundamental principles of symmetry of planar patterns. All can be used to effortlessly create periodic patterns . The educated user can exploit the programs to their fullest as educational and design tools. References 1. J.Britton and D. Seymour, Introduction to Tessellations (Palo Alto, CA: Dale Seymour, 1989). 2. Stan Wagon, Mathematica in Action (New York: Freeman, 1991) pp. 108-117. 3. For more information on this terminology, see Doris Schattschneider, "In Black and White: How to Create Perfectly Colored Symmetric Patterns ", Computers and Mathematics withApplications 12B, Nos 3/4, 673-695: (1986) and Symmetry: UnifyingHuman Understanding, Istvan Hargittai, ed. (Oxford: Pergamon, 1986) pp.673-695. 4. See Branko Griinbaum and Geoffrey Shephard, Tilingsand Patterns(New York: Freeman , 1987), especially pp. 288-290. DIFFERENTIAL AND DIFFERENCE EQUATIONS THROUGH COMPUTER EXPERIMENTS PHASER:AN ANIMATOR/SIMULATOR FOR DYNAMICAL SYSTEMS FOR IBM PERSONAL COMPUTERS by Hiiseyn Kocak, Springer-Verlag, New York, NY,U.S.A., 1986. 224 pp. ISBN: 0-387-96239-5. See review in the Book Reviews section. FIELDS & OPERATORS by Martin Lapidus, Lascaux Graphics, 3771 East Guthrie Mountain Place, Tucson, AZ 85718, U.S.A., 1989. MODELS by Martin Lapidus, Lascaux Graphics, 3771 East Guthrie Mountain Place, Tucson, AZ 85718, U.S.A., 1990. Reviewed byMicheleEmmer, Via Santa Maria della Speranza 5,00139 Rome, Italy. Fields & Operators is an interactive graphics program designed both to help users visualize surfaces, vector fields and integral flows, and to investigate the effects of differential operators on them. The program uses a system of 'layers' to display geometric figures. A layer can be a scalar field, a vector field or an integral flow. A single drawing consists ofup to six layers ; the calculations that define any' layer depends on the results from previous layers. Fields & Operators supports the following differential operators : gradient, divergence, curl, Laplacian, time derivative, second derivative . The user can save a sequence of frames to disk and later replay them to produce an animated picture . Even though the user cannot define new symbolic manipulation procedures and, in defining functions, no branching in the definition is allowed , the user interface is better than that in many other programs. With a mouse it is very easy to issue commands, and it is even amusing to use it. Fields & Operators is a good choice [or anyone interested in studying...