turano@silver.DEC (Tom Turano DTN 231- [Office 4735, Lab 6978]) (08/22/84)
* In regard to Glen Norris' question concerning fractals, the following reference may be of interest: "Geometrical forms known as fractals find sense in chaos", McDermott, SMITHSONIAN, December 1983, pg 110 (This is a graphically pleasing article.) In addition this paper states that the SCIENTIFIC AMERICAN Mathematical Games/Metamagical Themas sections of the December 1976, April 1978 and November 1981 issues were dedicated to fractals. Tom Turano Laboratory Data Products DIGITAL EQUIPMENT CORPORATION path: decvax!decwrl!rhea!gold!turano
sharp@aquila.UUCP (08/27/84)
I am surprised that noone has yet mentioned Mandelbrot's own book, the latest edition of which is called "The Fractal Geometry of Nature", W.H. Freeman, 1983. It is actually very clear and readable, with some excellent illustrations. It also has some discussion of chaos, although not fully up-to-date. [BTW, the most widely seen application of fractals must be the Genesis planet in the Star Trek movies II and III] -- Nigel Sharp [noao!sharp National Optical Astronomy Observatories]
wbp@hou2d.UUCP (W.PINEAULT) (08/27/84)
I am surprised that no mathematicians have yet responded to the question of what fractals are with a simple example or two, but I guess, Real Mathematicians don't read net ::-) (I defected to Computer Science) Forgive me if this reference was given before, but a very good introduction to fractals is: "The Fractal Geometry of Nature", by Benoit B. Mandelbrot, Freeman and CO, San Francisco. (The material here is paraphrased from this book.) The concept of fractals is actually a theoretical mathematical construction which seems to have implications for the real world. Part of the historical impetus for fractals comes from the concept of plane-filling curves (Peano curves), where a one dimensional line passes through every point on the plane, and so isn't its dimension really two? There are other historical roots, some are from real analysis dealing with Cantor sets and measures on them. The following is a concrete example: 1. Take a line of unit length: ______ 2. Replace the middle third by two lines each equal to a third of the unit: __/\__ 3. Repeat for each of the 4 new line segments with lines 1/3 of last produced lines. 4. Repeat for all new figures indefinitely. On the n'th application the length of the figure is (4/3)**n, and the length of the limit curve is therefor infinite (this takes a bit of proof, but is elementary calculus.) But saying that the length of this bounded curve, which mearly wiggles a lot, is infinite is not satisfying. Therefore a new concept was devised to somehow measure the unboundedness of this construction. Intuitive explanation: We measure the rate of groth of the series of curves defined above. Let L(n) be the length of the curve when using pieces of length n. Then we know: (1) L(n/3) = (4/3)L(n). Now we define the fractal dimension D of a curve to be that number which allows the following to be true: (2) L(n) = n**(1-D). In this case we substitute (2) into (1) and get: (n/3)**(1-D) = (4/3) (n**(1-D)) (1/3)**(1-D) = 4/3 (D-1) log(3) = log(4) - log (3) D = log(4) / log(3). Actual: The idea of fractal dimensions comes in part from the concept of Hausdorff measures. The essece of this idea is to take minamal coverings of a curve using disks of a fixed radius. Letting the radius approach 0, look at what happens to the sum of the diameters of the covering circles. Calculations are not hard and is a fairly interesting exercise. For each curve there is a unique number which makes the above work (the concept of fractional dimension is well defined), and for any number between 1 and 2 there exists a fractal with that dimension. Mandelbrot is the "father of fractals" and states that he thinks that they express some fundamental nature of the universe. His book certainly ties in enough disciplines from relativity to snowflakes. What is certain is that the concept of fractals takes care of many anomalies that have been troubling mathematicians for decades. Incidently only crves with a dimension not a unit are considered fractals. Wayne Pineault AT&T Consumer Products Holmdel, N.J.
schell@bnl.UUCP (Stephan) (08/28/84)
If anyone out there in net land has ever heard of fractals,
could someone please point me to some papers, journal articles,
etc. on the subject? THANKS in advance,
Stephan Schellschell@bnl.UUCP (Stephan) (08/29/84)
Please excuse the previous message -- I had failed to read the net.math stuff for a few days and noticed someone already gave the requested references.
mbh@epsilon.UUCP (Mark Hoffberg) (02/04/85)
Hey folks: I'm interested in finding an elementary (introductory, short) text or article on fractals. Please send mail. Thanks Mark Hoffberg Bellcore
kay@flame.UUCP (Kay Dekker) (02/15/85)
>Hey folks: > I'm interested in finding an elementary >(introductory, short) text or article on fractals. >Please send mail. > Thanks > Mark Hoffberg > Bellcore Me too, please! Kay. -- Ceci n'est pas une article. ... mcvax!ukc!ubu!flame!kay
lindley@ut-ngp.UUCP (John L. Templer) (02/17/85)
>Hey folks: > I'm interested in finding an elementary >(introductory, short) text or article on fractals. >Please send mail. > Thanks > Mark Hoffberg > Bellcore Me too, please! Kay. I sent Mark a response to his inquiry that might be of interest to you also. Two introductory books about fractals are: "Fractals: Form, Chance, and Dimension", and "The Fractal Geometry of Nature". Both are by Benoit Mandelbrot. And now, could someone recommend any other, more advanced, books on fractals? These two books are a little skimpy on the algorithms that Mandelbrot developed. -- John L. Templer University of Texas at Austin {allegra,gatech,seismo!ut-sally,vortex}!ut-ngp!lindley "Gongo Bunnies movin' in,
kastin@aecom.UUCP (Steven Kastin) (10/04/85)
I wonder if anyone can help me in my search for information on fractals. I already have the fractal "bible", "The Fractal Geometry of Nature" by Mandelbrot himself, but he assumes that those who read his book already know a lot about the subject. (for example the significance of omega.) I, unfortunately, do not, but was wondering if anyone can steer me toward some other reference material, overviews, etc. Any info will be greatly appreciated. Thanx. Steve Kastin, frustrated fractalist