Conference rusure::math

One interesting figure comes out when lambda real = 1.66 and imaginary = 1.11,
and scale factor = 400 (other values default).  Kinda like siamese dragons.

						John

I also played with 2-D fractals immediately after seeing that article...

If anyone has a PC350 running P/OS they ought to try my program; it
directly accesses the bitmap so must be orders of magnitude faster
than communicating via a comm line using REGIS.

I did a fair amount of playing around with the fractals and discovered
a number of things.

There are two basic starting points, from which the recurrance
Z(N+1) = LAMBDA*Z(N)*(1-Z(N)) loops... one is Z=0 and the other is
(LAMBDA-1)/LAMBDA.

The point Z=0 has a predecessor of Z=1. Plugging it in and solving the
quadratic gives a pair of roots - each of those has a pair of distinct
predecessors, and so on.  Exploring the resulting tree gives an estimate
of the outline of the fractal.  Always choosing one of the roots
(say the one with the positive sign) seems to always converge to a stable
set of points.  Thus I assume that always choosing one of the roots in some
pattern would converge to a stable set of points as well. I think this
convergence is what allows you to get an outline of the fractal by exploring
backwards as these programs do.

I wrote a pair of programs - one chose one of the roots at random and the
other explicitly searches the resulting tree to successivly greater
depths.

The task images are in ADVAX::USER1:[JROTH.BITMAP] FRACTR.TSK (random)
and FRACTL.TSK (tree exploration).  The souces are there as well.

There are a few other high speed graphics goodies for the PC350 there as well.
A few LIFE demos, a multi-color, multi-lobed cardioid demo (dig up a machine
with a color monitor for that one), a snowflake curve program, and a rotating
tesseract (four dimensional cube) program.

They all do direct access to the bitmap, since I've found it to be far far
faster than using GIDIS or core graphics.

Have fun -

	Jim

A comment about Jim Ravan's program...  since the points on any fractal
it produces are skew symmetric about the point (.5,0.) one should take
this into account by subtracting .5 before plotting to keep the pattern
centered. Also, you might as well plot both roots each time and then
choose one at random to proceed from.  Also, its better to start from
the point (1.,0.) instead of recursing backwards 10 times from the point
(.5,0.) since the point (1.,0.) is guaranteed to be on the curve... no
need to converge to it.

I changed the program slightly for this - its in the [jroth.bitmap]
directory.  The speed is not bad.  Good starting values of lambda are
in the neighborhood of magnitude .5 to 3 at any angle. Its interesting
to sweep lambda over a range and see how the curve crinkles up.

- Jim

Here are some technical references:

Benoit B. Mandelbrot, Self-Inverse Fractals Osculated by Sigma-Discs and the
	Limit Sets of Inversion Groups, The Mathematical Intelligencer.
	5(1983) no. 2, pp. 9-17.

Benoit B. Mandelbrot, The Fractal Geometry of Nature. W. H. Freeman.
	San Francisco: 1982.

Anthony Barcellos, The Fractal Geometry of Mandelbrot. The College
	Mathematics Journal. 15(1984)98-114.

Martin Gardner, Mathematical Games. Scientific American.
	Dec. 1976 and also April 1978.

William J. Gilbert, Geometry of Radix Representations. in The Geometric Vein,
	edited by Davis, Grunbaum and Sherk. Springer-Verlag. New York: 1981.
	pp. 129-139.

William J. Gilbert. The fractal dimension of snoflake spirals. Notices of the
	American Mathematical Society. 25(1978) A-641.

Chandler Davis and Donald Knuth, Number Representations and dragon curves,
	I, II. Journal of Recreational Mathematics. 3(1970) 66-81, 133-149.

Actually, there are (many) other "cycles" from Z   = L Z (1 - Z );
in fact, an infinite number of them.		n+1	n      n

				  L - 1
Z  = Z	has roots R  = 0, or R  = ----- (as noted).
 1    0		   0	      1	    L
					  2
			  L + 1 +/- sqrt(L  - 2 L - 3)
Z  = Z	has roots R ,R  = ----------------------------, R  and R  (as above).
 2    0	           2  3		    2 L			 0	1

Z  = Z  has roots R , and R  (as above), and four other roots.
 3    0		   0       1

Z  = Z  has roots R , R , R , and R  (as above), and four other roots.
 4    0		   0   1   2	   3

				  2
Setting L to one of the roots of L  - 2 L - 3 = 0 produces an interesting curve.

Newsgroups: net.math
Path: decwrl!decvax!mcnc!unc!ulysses!mhuxl!houxm!houxf!vax135!petsd!cjh
Subject: Fractals: recent articles
Posted: Thu Sep 13 16:24:42 1984

	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

re .2

Some of your 350 programs come up with "Common block not loaded". ??

q

Install PROF77 (the FORTRAN cluster library common)

- Jim