cjh@petsd.UUCP (Chris Henrich) (09/14/84)
[This space is blank.] Fractals appear in several articles in _The Mathematical Intelligencer_. This is a quarterly, published by Springer-Verlag. It has lots of good articles on mathematics and related subjects, at a level of difficulty higher than that of _Scientific American_ but not as high as a typical research paper. Another location for mathematical articles with lots of meaty content and general interest is the _Bulletin of the American Mathematical Society._ The Bulletin is addressed primarily to university mathematicians, and its articles assume the reader is familiar with modern math terminology and background. However, its survey articles are addressed to as "general" a public as meets that requirement. Here are some articles in which fractal sets, or sets whose boundaries are fractal, are constructed directly: (In my references, TMI = _The Mathematical Intelligencer_ and BAMS = _Bulletin of the AMS_.) Lenstra, Hendrik W., Jr. Euclidean number fields 3. TMI 2(1979-80)#2, 99-103. Gilbert, W. J. Fractal geometry derived from complex bases. TMI 4(1982)#2, 78-87. Mandelbrot, Benoit B. Self-inverse fractals osculated by sigma-disks and the limit sets of inversion groups. TMI 5(1983)#2, 9-17. Fractals appear in several ways in the study of dynamical systems. One kind of fractal set was studied as far back as 1919, by Fatou and Julia. An article with striking graphics is Peitgen, H. O., Saipe, D., Haeseler, F.v. Cayley's problem and Julia sets. TMI 6(1984)#2,11-20. A companion article is: Blanshard, Paul. Complex analytic dynamics on the Riemann sphere. BAMS 11 (1984)#1, 85-142. Fractals also appear in dynamical systems as "attractors." The terminology seems to be fluid here; fractal attractors were called "strange" until people got more used to fractal sets generally, and sometimes were called "chaotic". To get the straight skinny on these, I recommend starting with Hirsch, Morris W. The dynmical systems approach to differential equations, BAMS, 11 (1984)#1, 1-65 and then Ruelle, David. Strange Attractors. TMI 2(1979-1980)#4, 126-140. Fractal attractors challenge many assumptions we are likely to take for granted. When the trajectory of a dynamical system is attracted to a fractal attractor, it may appear to repeat its behavior in an vaguely periodic fashion, but the repetition never settles down to be quite predictable over a long time span. A system defined by simple equations, with behavior that (for a short time) is quite smooth and easily computable, may over a long time span produce a very good likeness of a "random" system, whatever that _really_ is. Regards, Chris Henrich Perkin-Elmer Computer Systems Division