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From: stepp@muvms6.mu.wvnet.edu (Ermel Stepp)
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Subject: Fractal Frequently Asked Questions and Answers
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Date: 22 Mar 1995 23:19:51 GMT
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Summary: Fractal images, software, algorithms, definitions, and reference
Keywords: fractals, chaos, complexity, Mandelbrot
X-Last-Updated: 1995/02/20
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Archive-name: fractal-faq  
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Last-modified: Mon, 13-Feb-1995
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FRACTAL FAQ (FREQUENTLY ASKED QUESTIONS)  
___________________________________________________________ 
ISSN Pending       Volume 1 Number 1      February 13, 1995 
___________________________________________________________ 
(c) Copyright Ermel Stepp 1995 
___________________________________________________________ 
  
Introduction  
  
The international computer network Usenet contains discussions on a  
variety of topics. The Usenet newsgroup sci.fractals and the listserv forum  
frac-l are devoted to discussions on fractals. This FAQ (Frequently Asked  
Questions) is an electronic serial compiled from questions and answers 
contributed by many participants in those discussions. This FAQ also 
lists various archives of programs, images, and papers that can be accessed 
through the global computer networks (WWW/Internet/BITNET) by using email, 
anonymous ftp, gophers, and World Wide Web browsers. This FAQ is not 
intended as a general introduction to fractals, or a set of rigorous 
definitions, but rather a useful summary of ideas, sources, and references.  
  
This FAQ is posted monthly to the Usenet groups sci.fractals, sci.answers,  
news.answers, bit.listserv.frac-l and the listserv forum frac-l. Like most
FAQs, it can be obtained free with a WWW browser or by anonymous ftp to 
ftp://rtfm.mit.edu/pub/usenet/news.answers/fractal-faq [18.181.0.24]; 
also, with a text-based browser, such as lynx, or anonymous ftp to: 
byrd.mu.wvnet.edu/pub/estepp/fracha/fractal.faq [129.71.32.152].  
It can be retrieved by email to mail-server@rtfm.mit.edu with the 
message: send usenet/news.answers/fractal-faq  
  
The hypertext version of the Fractal FAQ has hyperlinks to sources on the  
World Wide Web. It can be accessed with a browser such as xmosaic at  
http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html.  
Also, the hypertext version is online for review and comment at:
http://www.marshall.edu/~stepp/fractal-faq/faq.html.  
Please suggest other links to add to the Fractal FAQ.  
  
For your information, the World Wide Web FAQ is available via: 
  The WWW:  http://sunsite.unc.edu/boutell/faq/www_faq.html 
  Anonymous ftp:  rtfm.mit.edu in /pub/usenet/news.answers/www/faq 
  Email:  mail-server@rtfm.mit.edu (send usenet/news.answers/www/faq
  
If you are viewing this file with a newsreader such as "rn" or "trn", you can  
search for a particular question by using "g^Qn" (that's lower-case g, up-  
arrow, Q, and n, the number of the question you wish). Or you may  
browse forward using <control-G> to search for a Subject: line.  
  
I am happy to receive more information to add to this file. Also, let me  
know if you find mistakes. Please send your comments and suggestions  
to Ermel Stepp (email: stepp@marshall.edu).  
  
The questions which are answered are:  
Q1: I want to learn about fractals. What should I read first?  
Q2: What is a fractal? What are some examples of fractals?  
Q3: What is chaos?  
Q4a: What is fractal dimension? How is it calculated?  
Q4b: What is topological dimension?  
Q5: What is a strange attractor?  
Q6a: What is the Mandelbrot set?  
Q6b: How is the Mandelbrot set actually computed?  
Q6c: Why do you start with z=0?  
Q6d: What are the bounds of the Mandelbrot set? When does it diverge?  
Q6e: How can I speed up Mandelbrot set generation?  
Q6f: What is the area of the Mandelbrot set?  
Q6g: What can you say about the structure of the Mandelbrot set?  
Q6h: Is the Mandelbrot set connected?  
Q7a: What is the difference between the Mandelbrot set and a Julia set?  
Q7b: What is the connection between the Mandelbrot set and Julia sets?  
Q7c: How is a Julia set actually computed?  
Q7d: What are some Julia set facts?  
Q8a: How does complex arithmetic work?  
Q8b: How does quaternion arithmetic work?  
Q9: What is the logistic equation?  
Q10: What is Feigenbaum's constant?  
Q11a: What is an iterated function system (IFS)?  
Q11b: What is the state of fractal compression?  
Q12a: How can you make a chaotic oscillator?  
Q12b: What are laboratory demonstrations of chaos?  
Q13: What are L-systems?  
Q14: What is some information on fractal music?  
Q15: How are fractal mountains generated?  
Q16: What are plasma clouds?  
Q17a: Where are the popular periodically-forced Lyapunov fractals described?  
Q17b: What are Lyapunov exponents?  
Q17c: How can Lyapunov exponents be calculated?  
Q18: Where can I get fractal T-shirts and posters?  
Q19: How can I take photos of fractals?  
Q20: How can 3-D fractals be generated?  
Q21a: What is Fractint?  
Q21b: How does Fractint achieve its speed?  
Q22: Where can I obtain software packages to generate fractals?  
Q23a: How does anonymous ftp work?  
Q23b: What if I can't use ftp to access files?  
Q24a: Where are fractal pictures archived?  
Q24b: How do I view fractal pictures from alt.binaries.pictures.fractals?  
Q25: Where can I obtain fractal papers?  
Q26: How can I join the BITNET fractal discussion?  
Q27: What is complexity?  
Q28a: What are some general references on fractals and chaos?  
Q28b: What are some relevant journals?  
Q29: Are there any special notices?  
Q30: Who has contributed to the Fractal FAQ?  
Q31: Copyright?  
  
------------------------------  
  
Subject: Learning about fractals  
  
Q1: I want to learn about fractals. What should I read/view first?  
A1: _Chaos_ is a good book to get a general overview and history. _Fractals  
Everywhere_ is a textbook on fractals that describes what fractals are and  
how to generate them, but it requires knowing intermediate analysis.  
_Chaos, Fractals, and Dynamics_ is also a good start. There is a longer  
book list at the end of this file (see "What are some general references?").  
Also, use networked resources such as:  
  
http://millbrook.lib.rmit.edu.au/exploring.html Exploring Chaos and Fractals  
  
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html Fractal Microscope  
  
http://is.dal.ca:3400/~adiggins/fractal/ Dalhousie University Fractal Gallery  
  
http://acat.anu.edu.au/contours.html "Contours of the Mind"   
  
http://www.maths.tcd.ie/pub/images/images.html Computer Graphics Gallery  
  
http://wwfs.aist-na.ac.jp/shika/library/fractal/ SHiKA Fractal Image 
Library  
  
http://www.awa.com/sfff/sfff.html The San Francisco Fractal Factory.   
  
http://spanky.triumf.ca/www/spanky.html Spanky (Noel Giffin)  
  
http://www.cnam.fr/fractals.html Fractal Gallery (Frank Rousell)   
  
http://www.cnam.fr/fractals/anim.html Fractal Animations Gallery  
(Frank Rousell)  
  
------------------------------  
  
Subject: What is a fractal?  
  
Q2: What is a fractal? What are some examples of fractals?  
A2: A fractal is a rough or fragmented geometric shape that can be  
subdivided in parts, each of which is (at least approximately) a 
reduced-size copy of the whole. Fractals are generally self-similar 
and independent of scale.  
  
There are many mathematical structures that are fractals; e.g. Sierpinski  
triangle, Koch snowflake, Peano curve, Mandelbrot set, and Lorenz  
attractor. Fractals also describe many real-world objects, such as clouds,  
mountains, turbulence, and coastlines, that do not correspond to simple  
geometric shapes.  
  
Benoit Mandelbrot gives a mathematical definition of a fractal as a set for  
which the Hausdorff Besicovich dimension strictly exceeds the topological  
dimension. However, he is not satisfied with this definition as it excludes  
sets one would consider fractals.  
  
According to Mandelbrot, who invented the word: "I coined _fractal_ from  
the Latin adjective _fractus_. The corresponding Latin verb _frangere_  
means "to break:" to create irregular fragents. It is therefore sensible -  
and how appropriate for our needs! - that, in addition to "fragmented" (as in  
_fraction_ or _refraction_), _fractus_ should also mean "irregular," both  
meanings being preserved in _fragment_." (_The Fractal Geometry of  
Nature_, page 4.)  
  
------------------------------  
  
Subject: Chaos  
  
Q3: What is chaos?  
A3: Chaos is apparently unpredictable behavior arising in a deterministic  
system because of great sensitivity to initial conditions. Chaos arises in a  
dynamical system if two arbitrarily close starting points diverge exponential-  
ly, so that their future behavior is eventually unpredictable.  
  
Weather is considered chaotic since arbitrarily small variations in initial  
conditions can result in radically different weather later. This may limit  
the possibilities of long-term weather forecasting. (The canonical example  
is the possibility of a butterfly's sneeze affecting the weather enough to  
cause a hurricane weeks later.)  
  
Devaney defines a function as chaotic if it has sensitive dependence on ini-  
tial conditions, it is topologically transitive, and periodic points are  
dense. In other words, it is unpredictable, indecomposable, and yet contains  
regularity.  
  
Allgood and Yorke define chaos as a trajectory that is exponentially unstable  
and neither periodic or asymptotically periodic. That is, it oscillates ir-  
regularly without settling down.  
  
The following resources may be helpful to understand chaos:  
  
http://millbrook.lib.rmit.edu.au/exploring.html Exploring Chaos and Fractals  
  
http://www.cc.duth.gr/~mboudour/nonlin.html Chaos and Complexity  
Homepage (M. Bourdour)  
  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/lorenz.gif  
Lorenz attractor  
  
http://ucmp1.berkeley.edu/henon.html Experimental interactive  
henon attractor   
  
------------------------------  
  
Subject: Fractal dimension  
  
Q4a: What is fractal dimension? How is it calculated?  
A4a: A common type of fractal dimension is the Hausdorff-Besicovich  
Dimension, but there are several different ways of computing fractal  
dimension.  
  
Roughly, fractal dimension can be calculated by taking the limit of the quo-  
tient of the log change in object size and the log change in measurement  
scale, as the measurement scale approaches zero. The differences come in  
what is exactly meant by "object size" and what is meant by "measurement  
scale" and how to get an average number out of many different parts of a  
geometrical object. Fractal dimensions quantify the static *geometry* of an  
object.  
  
For example, consider a straight line. Now blow up the line by a factor of  
two. The line is now twice as long as before. Log 2 / Log 2 = 1,  
corresponding to dimension 1. Consider a square. Now blow up the square  
by a factor of two. The square is now 4 times as large as before (i.e. 4  
original squares can be placed on the original square). Log 4 / log 2 = 2,  
corresponding to dimension 2 for the square. Consider a snowflake curve  
formed by repeatedly replacing ___ with _/\_, where each of the 4 new lines  
is 1/3 the length of the old line. Blowing up the snowflake curve by a factor  
of 3 results in a snowflake curve 4 times as large (one of the old snowflake  
curves can be placed on each of the 4 segments _/\_).  
Log 4 / log 3 = 1.261... Since the dimension 1.261 is larger than the  
dimension 1 of the lines making up the curve, the snowflake curve is a  
fractal.  
  
For more information on fractal dimension and scale, access via the WWW  
http://life.anu.edu.au/complex_systems/tutorial3.html .  
  
Fractal dimension references:  
  
[1] J. P. Eckmann and D. Ruelle, _Reviews of Modern Physics_ 57, 3  
(1985), pp. 617-656.  
  
[2] K. J. Falconer, _The Geometry of Fractal Sets_, Cambridge Univ.  
Press, 1985.  
  
[3] T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for  
Chaotic Systems_, Springer Verlag, 1989.  
  
[4] H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,  
Springer-Verlag Inc., New York, 1988. ISBN 0-387-96608-0. This book  
contains many color and black and white photographs, high level math, and  
several pseudocoded algorithms.  
  
[5] G. Procaccia, _Physica D_ 9 (1983), pp. 189-208.  
  
[6] J. Theiler, _Physical Review A_ 41 (1990), pp. 3038-3051.  
  
References on how to estimate fractal dimension:  
  
1. S. Jaggi, D. A. Quattrochi and N. S. Lam, Implementation and  
operation of three fractal measurement algorithms for analysis of remote-  
sensing data., _Computers & Geosciences_ 19, 6 (July 1993), pp. 745-767.  
  
2. E. Peters, _Chaos and Order in the Capital Markets_, New York, 1991.  
ISBN 0-471-53372-6 Discusses methods of computing fractal dimension.   
Includes several short programs for nonlinear analysis.  
  
3. J. Theiler, Estimating Fractal Dimension, _Journal of the Optical Society  
of America A-Optics and Image Science_ 7, 6 (June 1990), pp. 1055-1073.  
  
There are some programs available to compute fractal dimension. They are  
listed in a section below (see "Fractal software").  
  
Q4b: What is topological dimension?  
A4b: Topological dimension is the "normal" idea of dimension; a point has  
topological dimension 0, a line has topological dimension 1, a surface has  
topological dimension 2, etc.  
  
For a rigorous definition:  
  
A set has topological dimension 0 if every point has arbitrarily small  
neighborhoods whose boundaries do not intersect the set.  
  
A set S has topological dimension k if each point in S has arbitrarily small  
neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the  
least nonnegative integer for which this holds.  
  
------------------------------  
  
Subject: Strange attractors  
  
Q5: What is a strange attractor?  
A5: A strange attractor is the limit set of a chaotic trajectory. A strange  
attractor is an attractor that is topologically distinct from a periodic orbit  
or a limit cycle. A strange attractor can be considered a fractal attractor.  
An example of a strange attractor is the Henon attractor.  
  
Consider a volume in phase space defined by all the initial conditions a  
system may have. For a dissipative system, this volume will shrink as the  
system evolves in time (Liouville's Theorem). If the system is sensitive to  
initial conditions, the trajectories of the points defining initial 
conditions will move apart in some directions, closer in others, but 
there will be a net shrinkage in volume. Ultimately, all points will 
lie along a fine line of zero volume. This is the strange attractor.  All 
initial points in phase space which ultimately land on the attractor 
form a Basin of Attraction. A strange attractor results if a system is
sensitive to initial conditions and is not conservative.  
  
Note: While all chaotic attractors are strange, not all strange attractors 
are chaotic. Reference:  
  
1. Grebogi, et al., Strange Attractors that are not Chaotic, _Physica D_ 13  
(1984), pp. 261-268.  
  
------------------------------  
  
Subject: The Mandelbrot set  
  
Q6a: What is the Mandelbrot set?  
A6a: The Mandelbrot set is the set of all complex c such that iterating
z -> z^2+c does not go to infinity (starting with z=0).  
  
An image of the Mandelbrot set is available on the WWW at  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/mandel1.gif .  
  
Other images and resources are:  
  
Frank Rousells two hyperindex of clickable/retrievable Mandelbrot images:  
ftp://ftp.cnam.fr/pub/Fractals/mandel/Index.gif Mandelbrot Images  
(Frank Rousell)  
ftp://ftp.cnam.fr/pub/Fractals/mandel/Index2.gif Mandebrot Images #2
(Frank Rousell)  
  
http://www.wpl.erl.gov/misc/mandel.html Interactive Mandelbrot  
(Neal Kettler)   
  
http://www.ntua.gr/mandel/mandel.html Mandelbrot Explorer (interactive)  
(Panagiotis J. Christias)   
  
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html  
Fractal Microscope   
  
http://hermes.cybernetics.net/distfract.html Distributed Fractal Generator  
for SunOS Sparcstations (James Robinson)   
  
Q6b: How is the Mandelbrot set actually computed?  
A6b: The basic algorithm is:  
For each pixel c, start with z=0. Repeat z=z^2+c up to N times, exiting if  
the magnitude of z gets large.  
If you finish the loop, the point is probably inside the Mandelbrot set. If  
you exit, the point is outside and can be colored according to how many  
iterations were completed. You can exit if |z|>2, since if z gets this big it  
will go to infinity. The maximum number of iterations, N, can be selected  
as desired, for instance 100. Larger N will give sharper detail but take  
longer.  
  
Q6c: Why do you start with z=0?  
A6c: Zero is the critical point of z^2+c, that is, a point where 
d/dz (z^2+c) = 0. If you replace z^2+c with a different function, the 
starting value will have to be modified. E.g. for z->z^2+z+c, the 
critical point is given by 2z+1=0, so start with z=-1/2. In some cases, 
there may be multiple critical values, so they all should be tested.  
  
Critical points are important because by a result of Fatou: every attracting  
cycle for a polynomial or rational function attracts at least one critical  
point. Thus, testing the critical point shows if there is any stable  
attractive cycle. See also:  
  
1. M. Frame and J. Robertson, A Generalized Mandelbrot Set and the  
Role of Critical Points, _Computers and Graphics_ 16, 1 (1992), pp. 35-40.  
  
Note that you can precompute the first Mandelbrot iteration by starting with  
z=c instead of z=0, since 0^2+c=c.  
  
Q6d: What are the bounds of the Mandelbrot set? When does it diverge?  
A6d: The Mandelbrot set lies within |c|<=2. If |z| exceeds 2, the z sequence  
diverges. Proof: if |z|>2, then |z^2+c|>= |z^2|-|c|> 2|z|-|c|. If  
|z|>=|c|, then 2|z|-|c|> |z|. So, if |z|>2 and |z|>=c, |z^2+c|>|z|, so the  
sequence is increasing. (It takes a bit more work to prove it is unbounded  
and diverges.) Also, note that |z1=c, so if |c|>2, the sequence diverges.  
  
Q6e: How can I speed up Mandelbrot set generation?  
A6e: See the information on speed below (see "Fractint"). Also see:  
  
1. R. Rojas, A Tutorial on Efficient Computer Graphic Representations of the  
Mandelbrot Set, _Computers and Graphics_ 15, 1 (1991), pp. 91-100.  
  
Q6f: What is the area of the Mandelbrot set?  
A6f: Ewing and Schober computed an area estimate using 240,000 terms of the  
Laurent series. The result is 1.7274... However, the Laurent series  
converges very slowly, so this is a poor estimate. A project to measure the  
area via counting pixels on a very dense grid shows an area around 1.5066.  
(Contact mrob@world.std.com for more information.) Hill and Fisher used  
distance estimation techniques to rigorously bound the area and found 
the area is between 1.503 and 1.5701.  
  
References:  
  
1. J. H. Ewing and G. Schober, The Area of the Mandelbrot Set, _Numer.  
Math._ 61 (1992), pp. 59-72.  
  
2. Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot Set,  
_Numerische Mathematik_, . (Submitted for publication). Available by  
ftp: legendre.ucsd.edu:/pub/Research/Fischer/area.ps.Z ..  
  
Q6g: What can you say about the structure of the Mandelbrot set?  
A6g: Most of what you could want to know is in Branner's article in _Chaos  
and Fractals: The Mathematics Behind the Computer Graphics_.  
  
Note that the Mandelbrot set in general is _not_ strictly self-similar; the  
tiny copies of the Mandelbrot set are all slightly different, mainly because  
of the thin threads connecting them to the main body of the Mandelbrot set.  
However, the Mandelbrot set is quasi-self-similar. The Mandelbrot set is  
self-similar under magnification in neighborhoods of Misiurewicz points,  
however (e.g. -.1011+.9563i). The Mandelbrot set is conjectured to be  
self- similar around generalized Feigenbaum points (e.g. -1.401155 or  
-.1528+1.0397i), in the sense of converging to a limit set. References:  
  
1. T. Lei, Similarity between the Mandelbrot set and Julia Sets,  
_Communications in Mathematical Physics_ 134 (1990), pp. 587-617.  
  
2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in  
_Computers in Geometry and Topology_, M. Tangora (editor), Dekker,  
New York, pp. 211-257.  
  
The "external angles" of the Mandelbrot set (see Douady and Hubbard or  
brief sketch in "Beauty of Fractals") induce a Fibonacci partition onto it.  
  
The boundary of the Mandelbrot set and the Julia set of a generic c in M  
have Hausdorff dimension 2 and have topological dimension 1. The proof  
is based on the study of the bifurcation of parabolic periodic points. (Since  
the boundary has empty interior, the topological dimension is less than 2,  
and thus is 1.) Reference:  
  
1. M. Shishikura, The Hausdorff Dimension of the Boundary of the  
Mandelbrot Set and Julia Sets, The paper is available from anonymous ftp:  
math.sunysb.edu:/preprints/ims91-7.ps.Z [129.49.18.1]..  
  
Q6h: Is the Mandelbrot set connected?  
A6h: The Mandelbrot set is simply connected. This follows from a theorem  
of Douady and Hubbard that there is a conformal isomorphism from the  
complement of the Mandelbrot set to the complement of the unit disk. (In  
other words, all equipotential curves are simple closed curves.) It is  
conjectured that the Mandelbrot set is locally connected, and thus pathwise  
connected, but this is currently unproved.  
  
Connectedness definitions:  
  
Connected: X is connected if there are no proper closed subsets A and B of  
X such that A union B = X, but A intersect B is empty. I.e. X is connected  
if it is a single piece.  
  
Simply connected: X is simply connected if it is connected and every closed  
curve in X can be deformed in X to some constant closed curve. I.e. X is  
simply connected if it has no holes.  
  
Locally connected: X is locally connected if for every point p in X, for  
every open set U containing p, there is an open set V containing p and  
contained in the connected component of p in U. I.e. X is locally connected  
if every connected component of every open subset is open in X.  
  
Arcwise (or path) connected: X is arcwise connected if every two points in  
X are joined by an arc in X.  
  
(The definitions are from _Encyclopedic Dictionary of Mathematics_.)  
  
------------------------------  
  
Subject: Julia sets  
  
Q7a: What is the difference between the Mandelbrot set and a Julia set?  
A7a: The Mandelbrot set iterates z^2+c with z starting at 0 and varying c.  
The Julia set iterates z^2+c for fixed c and varying starting z values. That  
is, the Mandelbrot set is in parameter space (c-plane) while the Julia set is  
in dynamical or variable space (z-plane).  
  
Q7b: What is the connection between the Mandelbrot set and Julia sets?  
A7b: Each point c in the Mandelbrot set specifies the geometric structure of  
the corresponding Julia set. If c is in the Mandelbrot set, the Julia set  
will be connected. If c is not in the Mandelbrot set, the Julia set will be a  
Cantor dust.  
  
You can see an example Julia set on the WWW at  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/julia.gif .  
  
Q7c: How is a Julia set actually computed?  
A7c: The Julia set can be computed by iteration similar to the Mandelbrot  
computation. The only difference is that the c value is fixed and the 
initial z value varies.  
  
Alternatively, points on the boundary of the Julia set can be computed 
quickly by using inverse iterations. This technique is particularly 
useful when the Julia set is a Cantor Set. In inverse iteration, the
equation z1 = z0^2+c is reversed to give an equation for
z0: z0 = +- sqrt(z1-c). By applying this equation repeatedly, the 
resulting points quickly converge to the Julia set boundary. (At each 
step, either the postive or negative root is randomly selected.) This
is a nonlinear iterated function system. In pseudocode: z = 1 (or any
value) loop  
 if (random number < .5) then  
  z = sqrt(z-c)  
 else  
  z =-sqrt(z-c)  
 endif  
 plot z  
end loop  
  
Q7d: What are some Julia set facts?  
A7d: The Julia set of any rational map of degree greater than one is perfect  
(hence in particular uncountable and nonempty), completely invariant, equal  
to the Julia set of any iterate of the function, and also is the boundary
of the basin of attraction of every attractor for the map.  
  
Julia set references:  
  
1. A. F. Beardon, _Iteration of Rational Functions : Complex Analytic  
Dynamical Systems_, Springer-Verlag, New York, 1991.  
  
2. P. Blanchard, Complex Analytic Dynamics on the Riemann Sphere, _Bull. of  
the Amer. Math. Soc_ 11, 1 (July 1984), pp. 85-141. This article is a  
detailed discussion of the mathematics of iterated complex functions. It  
covers most things about Julia sets of rational polynomial functions.  
  
------------------------------  
  
Subject: Complex arithmetic and quaternion arithmetic  
  
Q8a: How does complex arithmetic work?  
A8a: It works mostly like regular algebra with a couple additional formulas:  
(note: a,b are reals, x,y are complex, i is the square root of -1)  
Powers of i: i^2 = -1  
Addition: (a+i*b)+(c+i*d) = (a+c)+i*(b+d)  
Multiplication: (a+i*b)*(c+i*d) = a*c-b*d + i*(a*d+b*c)  
Division: (a+i*b)/(c+i*d) = (a+i*b)*(c-i*d)/(c^2+d^2)  
Exponentiation: exp(a+i*b) = exp(a)(cos(b)+i*sin(b))  
Sine: sin(x) = (exp(i*x)-exp(-i*x))/(2*i)  
Cosine: cos(x) = (exp(i*x)+exp(-i*x))/2  
Magnitude: |a+i*b= sqrt(a^2+b^2)  
Log: log(a+i*b) = log(|a+i*b|)+i*arctan(b/a) (Note: log is multivalued.)  
Log (polar coordinates): log(r*e^(i*theta)) = log(r)+i*theta  
Complex powers: x^y = exp(y*log(x))  
DeMoivre's theorem: x^a = r^a * [cos(a*theta) + i * sin(a*theta)]  
More details can be found in any complex analysis book.  
  
Q8b: How does quaternion arithmetic work?  
A8b: Quaternions have 4 components (a+ib+jc+kd) compared to the two of  
complex numbers. Operations such as addition and multiplication can be  
performed on quaternions, but multiplication is not commutative..  
Quaternions satisfy the rules i^2=j^2=k^2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j.  
  
See:  
  
http://www.dtek.chalmers.se/Datorsys/Project/qjulia/index.html   
QJulia page (quaternions) (Henrik Engstrvm)   
  
------------------------------  
  
Subject: Logistic equation  
  
Q9: What is the logistic equation?  
A9: It models animal populations. The equation is x -> c*x*(1-x), where x  
is the population (between 0 and 1) and c is a growth constant. Iteration of  
this equation yields the period doubling route to chaos. For c between  
1 and 3, the population will settle to a fixed value. At 3, the period 
doubles to 2; one year the population is very high, causing a low population
the next year, causing a high population the following year. At 3.45, the
period  doubles again to 4, meaning the population has a four year cycle. 
The period keeps doubling, faster and faster, at 3.54, 3.564, 3.569, and 
so forth.  At 3.57, chaos occurs; the population never settles to a fixed 
period. For most c values between 3.57 and 4, the population is chaotic, 
but there are also periodic regions. For any fixed period, there is some 
c value that will yield that period. See "An Introduction to Chaotic 
Dynamical Systems" for more information.  
  
------------------------------  
  
Subject: Feigenbaum's constant  
  
Q10: What is Feigenbaum's constant?  
A10: In a period doubling cascade, such as the logistic equation, consider  
the parameter values where period-doubling events occur (e.g.  
r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of distances
between consecutive doubling parameter values; let  
delta[n] = (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to 
infinity is Feigenbaum's (delta) constant.  
  
Based on independent computations by Jay Hill and Keith Briggs, it has the  
value 4.669201609102990671853... Note: several books have published  
incorrect values starting 4.66920166...; the last repeated 6 is a 
typographical error.  
  
The interpretation of the delta constant is as you approach chaos, each  
periodic region is smaller than the previous by a factor approaching 4.669...  
Feigenbaum's constant is important because it is the same for any function  
or system that follows the period-doubling route to chaos and has a one-  
hump quadratic maximum. For cubic, quartic, etc. there are different  
Feigenbaum constants.  
  
Feigenbaum's alpha constant is not as well known; it has the value  
2.502907875095. This constant is the scaling factor between x values at  
bifurcations. Feigenbaum says, "Asymptotically, the separation of adjacent  
elements of period-doubled attractors is reduced by a constant value [alpha]  
from one doubling to the next". If d[n] is the algebraic distance between  
nearest elements of the attractor cycle of period 2^n, then d[n]/d[n+1]  
converges to -alpha.  
  
References:  
  
1. K. Briggs, How to calculate the Feigenbaum constants on your PC,  
_Aust. Math. Soc. Gazette_ 16 (1989), p. 89.  
  
2. K. Briggs, A precise calculation of the Feigenbaum constants,  
_Mathematics of Computation_ 57 (1991), pp. 435-439.  
  
3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for  
Mandelsets, _J. Phys._ A24 (1991), pp. 3363-3368.  
  
4. M. Feigenbaum, The Universal Metric Properties of Nonlinear  
Transformations, _J. Stat. Phys_ 21 (1979), p. 69.  
  
5. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los  
Alamos Sci_ 1 (1980), pp. 1-4. Reprinted in _Universality in Chaos_ ,  
compiled by P. Cvitanovic.  
  
------------------------------  
  
Subject: Iterated function systems and compression  
  
Q11a: What is an iterated function system (IFS)?  
A11a: If a fractal is self-similar, you can specify mappings that map the  
whole onto the parts. Iteration of these mappings will result in convergence  
to the fractal attractor. An IFS consists of a collection of these (usually  
affine) mappings. If a fractal can be described by a small number of  
mappings, the IFS is a very compact description of the fractal. An iterated  
function system is By taking a point and repeatedly applying these mappings  
you end up with a collection of points on the fractal. In other words,  
instead of a single mapping x -> F(x), there is a collection of (usually  
affine) mappings, and random selection chooses which mapping is used.  
  
For instance, the Sierpinski triangle can be decomposed into three self-  
similar subtriangles. The three contractive mappings from the full triangle  
onto the subtriangles forms an IFS. These mappings will be of the form  
"shrink by half and move to the top, left, or right".  
  
Iterated function systems can be used to make things such as fractal ferns  
and trees and are also used in fractal image compression. _Fractals  
Everywhere_ by Barnsley is mostly about iterated function systems.  
  
The simplest algorithm to display an IFS is to pick a starting point,  
randomly select one of the mappings, apply it to generate a new point, plot  
the new point, and repeat with the new point. The displayed points will  
rapidly converge to the attractor of the IFS.  
  
An IFS fractal fern can be viewed on the WWW at  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/fern.gif .  
  
Frank Rousells hyperindex of clickable/retrievable IFS images:  
ftp://ftp.cnam.fr/pub/Fractals/ifs/Index.gif  
  
Q11b: What is the state of fractal compression?  
A11b: Fractal compression is quite controversial, with some people claiming  
it doesn't work well, and others claiming it works wonderfully. The basic  
idea behind fractal image compression is to express the image as an iterated  
function system (IFS). The image can then be displayed quickly and  
zooming will generate infinite levels of (synthetic) fractal detail. The  
problem is how to efficiently generate the IFS from the image.  
  
Barnsley, who invented fractal image compression, has a patent on fractal  
compression techniques (4,941,193). Barnsley's company, Iterated Systems  
Inc, has a line of products including a Windows viewer, compressor,  
magnifier program, and hardware assist board.  
  
Fractal compression is covered in detail in the comp.compression FAQ file  
(See "compression-FAQ"). Ftp: rtfm.mit.edu:/pub/usenet/comp.compression  
[18.181.0.24].  
  
Three books describing fractal image compression are:  
  
1. M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988. ISBN 0-  
12-079062-9. This is an excellent text book on fractals. This is probably  
the best book for learning about the math underpinning fractals. It is also a  
good source for new fractal types.  
  
2. M. Barnsley and L. Hurd, _Fractal Image Compression_, Jones and  
Bartlett. ISBN 0-86720-457-5. This book explores the science of the fractal  
transform in depth. The authors begin with a foundation in information  
theory and present the technical background for fractal image compression.  
In so doing, they explain the detailed workings of the fractal transform.  
Algorithms are illustrated using source code in C.  
.  
3. Y. Fisher (Ed), _Fractal Image Compression: Theory and Application_.  
Springer Verlag, 1995.  
  
The October 1993 issue of Byte discussed fractal compression. You can ftp  
sample code: ftp.uu.net:/published/byte/93oct/fractal.exe .  
  
An introductory paper is:  
  
1. A. E. Jacquin, Image Coding Based on a Fractal Theory of Iterated  
Contractive Image Transformation, _IEEE Transactions on Image  
Processing_, January 1992.  
  
A fractal decompression demo program is available by anonymous ftp:  
lyapunov.ucsd.edu:/pub/inls-ucsd/fractal-2.0 [132.239.86.10].  
  
Another MS-DOS compression demonstration program is available by  
anonymous ftp: lyapunov.ucsd.edu:/pub/young-fractal .  
  
A site with information on fractal compression is  
legendre.ucsd.edu:/pub/Research/Fisher . On the WWW you can access  
file://legendre.ucsd.edu/pub/Research/Fisher/fractal.html .  
  
Many fractal image compression papers are available from  
ftp.informatik.uni-freiburg.de:/documents/papers/fractal [132.230.150.1].  
A review of the literature is in Guide.ps.gz. See the README 
file for an overview of the available documents.  
  
Other references:  
  
http://dip1.ee.uct.ac.za/fractal.bib.html "Fractal Compression  
Bibliography"   
  
http://inls.ucsd.edu/y/Fractals/ Fractal Compression (Yuval Fisher )   
  
------------------------------  
  
Subject: Chaotic demonstrations  
  
Q12a: How can you make a chaotic oscillator?  
A12a: Two references are:  
  
1. T. S. Parker and L. O. Chua, Chaos: a tutorial for engineers,  
_Proceedings IEEE_ 75 (1987), pp. 982-1008.  
  
2. _New Scientist_, June 30, 1990, p. 37.  
  
Q12b: What are laboratory demonstrations of chaos?  
A12b: Robert Shaw at UC Santa Cruz experimented with chaos in dripping  
taps. This is described in:  
  
1. J. P. Crutchfield, Chaos, _Scientific American_ 255, 6 (Dec. 1986), pp.  
38-49.  
  
2. I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_,  
B. Blackwell, New York, 1989.  
  
Two references to other laboratory demonstrations are:  
  
1. K. Briggs, Simple Experiments in Chaotic Dynamics, _American Journal  
of Physics_ 55, 12 (Dec 1987), pp. 1083-1089.  
  
2. J. L. Snider, Simple Demonstration of Coupled Oscillations, _American  
Journal of Physics_ 56, 3 (Mar 1988), p. 200.  
  
------------------------------  
  
Subject: L-Systems  
  
Q13: What are L-systems?  
A13: A L-system or Lindenmayer system is a formal grammar for  
generating strings. (That is, it is a collection of rules such as replace
X with XYX.) By recursively applying the rules of the L-system to an 
initial string, a string with fractal structure can be created. Interpreting
this string as a set of graphical commands allows the fractal to be displayed.
L-systems are very useful for generating realistic plant structures.  
  
Some references are:  
  
1. P. Prusinkiewicz and J. Hanan, _Lindenmayer Systems, Fractals, and  
Plants_, Springer-Verlag, New York, 1989.  
  
2. P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of  
Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good  
book on L-systems, which can be used to model plants in a very realistic  
fashion. The book contains many pictures.  
  
More information can be obtained via the WWW at:  
  
http://life.anu.edu.au/complex_systems/tutorial2.html Tutorial  
  
gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/leaf.gif L-system  
leaf  
  
http://hill.lut.ac.uk:80/TestStuff/trees/ 3 Dim. L-system Tree program  
(P.J.Drinkwater)   
  
http://www.geom.umn.edu/pix/archive/subjects/L-systems.html  L-system  
images.   
  
------------------------------  
  
Subject: Fractal music  
  
Q14: What is some information on fractal music?  
A14: One fractal recording is "The Devil's Staircase: Composers and  
Chaos" on the Soundprint label.  
  
Some references, many from an unpublished article by Stephanie Mason,  
are:  
  
1. R. Bidlack, Chaotic Systems as Simple (But Complex) Compositional  
Algorithms, _Computer Music Journal_, Fall 1992.  
  
2. C. Dodge, A Musical Fractal, _Computer Music Journal_ 12, 13 (Fall  
1988), p. 10.  
  
3. K. J. Hsu and A. Hsu, Fractal Geometry of Music, _Proceedings of the  
National Academy of Science, USA_ 87 (1990), pp. 938-941.  
  
4. K. J. Hsu and A. Hsu, Self-similatrity of the '1/f noise' called music.,  
_Proceedings of the National Academy of Science USA_ 88 (1991), pp.  
3507-3509.  
  
5. C. Pickover, _Mazes for the Mind: Computers and the Unexpected_, St.  
Martin's Press, New York, 1992.  
  
6. P. Prusinkiewicz, Score Generation with L-Systems, _International  
Computer Music Conference 86 Proceedings_, 1986, pp. 455-457.  
  
7. _Byte_ 11, 6 (June 1986), pp. 185-196.  
  
An IBM-PC program for fractal music is available at  
ftp://spanky.triumf.ca in [pub.fractals.programs.ibmpc] WTF23.ZIP.  
[142.90.112.1]  
  
A fractal music C++ package is available at  
http://neural.hampshire.edu:10001/~gzenie/inSanity.html .  
  
Also, it may b helpful to access:  
  
http://www-ks.rus.uni-stuttgart.de/people/schulz/fmusic   
The Fractal Music Project (Claus-Dieter Schulz)   
  
http://www.ccsr.uiuc.edu/People/gmk/Projects/ChuaSoundMusic/ChuaSoundMusic.html  
Chua's Oscillator: Applications of Chaos to Sound and Music   
  
------------------------------  
  
Subject: Fractal mountains  
  
Q15: How are fractal mountains generated?  
A15: Usually by a method such as taking a triangle, dividing it into 3  
subtriangles, and perturbing the center point. This process is then repeated  
on the subtriangles. This results in a 2-d table of heights, which can then  
be rendered as a 3-d image. One reference is:  
  
1. M. Ausloos, _Proc. R. Soc. Lond. A_ 400 (1985), pp. 331-350.  
  
------------------------------  
  
Subject: Plasma clouds  
  
Q16: What are plasma clouds?  
A16: They are a Fractint fractal and are similar to fractal mountains.  
Instead of a 2-d table of heights, the result is a 2-d table of intensities.  
They are formed by repeatedly subdividing squares.  
  
Network resources:  
  
http://climate.gsfc.nasa.gov/~cahalan/FractalClouds/FractalClouds.html   
Fractal Clouds Reference (calahan@clouds.gsfc.nasa.gov)   
  
http://ivory.nosc.mil/html/trancv/html/cloud-fract.html   
Fractal generated clouds (cahalan@clouds.gsfc.nasa.gov)   
  
------------------------------  
  
Subject: Lyapunov fractals  
  
Q17a: Where are the popular periodically-forced Lyapunov fractals described?  
A17a: See:  
  
1. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,  
Sept.  
1991, pp. 178-180.  
  
2. M. Markus and B. Hess, Lyapunov Exponents of the Logistic Map with  
Periodic Forcing, _Computers and Graphics_ 13, 4 (1989), pp. 553-558.  
  
3. M. Markus, Chaos in Maps with Continuous and Discontinuous  
Maxima, _Computers in Physics_, Sep/Oct 1990, pp. 481-493.  
  
Q17b: What are Lyapunov exponents?  
A17b:  
  
Lyapunov exponents quantify the amount of linear stability or instability of  
an attractor, or an asymptotically long orbit of a dynamical system. There  
are as many lyapunov exponents as there are dimensions in the state space  
of the system, but the largest is usually the most important.  
  
Given two initial conditions for a chaotic system, a and b, which are close  
together, the average values obtained in successive iterations for a and b  
will differ by an exponentially increasing amount. In other words, the two  
sets of numbers drift apart exponentially. If this is written e^(n*(lambda))  
for n iterations, then e^(lambda) is the factor by which the distance between  
closely related points becomes stretched or contracted in one iteration.  
Lambda is the Lyapunov exponent. At least one Lyapunov exponent must  
be positive in a chaotic system. A simple derivation is available in:  
  
1. H. G. Schuster, _Deterministic Chaos: An Introduction_, Physics  
Verlag, 1984.  
  
Q17c: How can Lyapunov exponents be calculated?  
A17c: For the common periodic forcing pictures, the lyapunov exponent is:  
lambda = limit as N->infinity of 1/N times sum from n=1 to N of  
log2(abs(dx sub n+1 over dx sub n))  
  
In other words, at each point in the sequence, the derivative of the 
iterated equation is evaluated. The Lyapunov exponent is the average 
value of the log of the derivative. If the value is negative, the iteration
is stable. Note that summing the logs corresponds to multiplying the
derivatives; if the product of the derivatives has magnitude < 1, points
will get pulled closer together as they go through the iteration.  
  
MS-DOS and Unix programs for estimating Lyapunov exponents from  
short time series are available by ftp: lyapunov.ucsd.edu:/pub/ncsu .  
  
Computing Lyapunov exponents in general is more difficult. Some  
references are:  
  
1. H. D. I. Abarbanel, R. Brown and M. B. Kennel, Lyapunov Exponents  
in Chaotic Systems: Their importance and their evaluation using observed  
data, _International Journal of Modern Physics B_ 56, 9 (1991), pp. 1347-  
1375.  
  
2. A. K. Dewdney, Leaping into Lyapunov Space, _Scientific American_,  
Sept. 1991, pp. 178-180.  
  
3. M. Frank and T. Stenges, _Journal of Economic Surveys_ 2 (1988), pp.  
103- 133.  
  
4. T. S. Parker and L. O. Chua, _Practical Numerical Algorithms for  
Chaotic Systems_, Springer Verlag, 1989.  
  
------------------------------  
  
Subject: Fractal items  
  
Q18: Where can I get fractal T-shirts and posters?  
A18: One source is Art Matrix, P.O. box 880, Ithaca, New York, 14851, 1-  
800- PAX-DUTY. Another source is Media Magic; they sell many fractal  
posters, calendars, videos, software, t-shirts, ties, and a huge variety of  
books on fractals, chaos, graphics, etc. Media Magic is at PO Box 598  
Nicasio, CA 94946, 415-662-2426. A third source is Ultimate Image; they  
sell fractal t- shirts, posters, gift cards, and stickers. Ultimate Image
is at PO Box 7464, Nashua, NH 03060-7464. Another source is Dave Kliman  
(516)-625-1915, whose products are distributed through Spencer Gifts,  
Posterservice, 800 666 7654, and Scandecor International., and this spring,  
through JC Penny, featuring all-over fractal t-shirts. Cyber Fiber produces  
fractal silk scarves, t-shirts, and postcards. Contact Robin Lowenthal, Cyber  
Fiber, 4820 Gallatin Way, San Diego, CA 92117.  
  
------------------------------  
  
Subject: How can I take photos of fractals?  
  
Q19: How can I take photos of fractals?  
A19: Noel Giffin gets good results with the following setup:  
Use 100 asa Kodak gold for prints or 64 asa for slides.  
Use a long lens (100mm) to flatten out the field of view and minimize  
screen curvature.  
Use f4 stop. Shutter speed must be longer than frame rate to get a complete  
image; 1/4 seconds works well.  
Use a tripod and cable release or timer to get a stable picture. The room  
should be completely blackened, with no light, to prevent glare and to  
prevent the monitor from showing up in the picture.  
  
You can also obtain high quality images by sending your targa or gif images  
to a commercial graphics imaging shop. They can provide much higher  
resolution images. Prices are about $10 for a 35mm slide or negative and  
about $50 for a high quality 4x5 negative.  
  
------------------------------  
  
Subject: 3-D fractals  
  
Q20: How can 3-D fractals be generated?  
A20: A common source for 3-D fractals is to compute Julia sets with  
quaternions instead of complex numbers. The resulting Julia set is four  
dimensional. By taking a slice through the 4-D Julia set (e.g. by fixing one  
of the coordinates), a 3-D object is obtained. This object can then be  
displayed using computer graphics techniques such as ray tracing.  
  
View Frank Rousells hyperindex of clickable/retrievable 3D images:  
ftp://ftp.cnam.fr/pub/Fractals/3D/Index.gif  
  
The papers to read on this are:  
  
1. J. Hart, D. Sandin and L. Kauffman, Ray Tracing Deterministic 3-D  
Fractals, _SIGGRAPH_, 1989, pp. 289-296.  
  
2. A. Norton, Generation and Display of Geometric Fractals in 3-D,  
_SIGGRAPH_, 1982, pp. 61-67.  
  
3. A. Norton, Julia Sets in the Quaternions, _Computers and Graphics,_  
13, 2 (1989), pp. 267-278. Two papers on cubic polynomials, which can  
be used to generate 4-D fractals:  
  
1. B. Branner and J. Hubbard, The iteration of cubic polynomials, part I.,  
_Acta Math_ 66 (1988), pp. 143-206.  
  
2. J. Milnor, Remarks on iterated cubic maps, This paper is available from  
anonymous ftp: math.sunysb.edu:/preprints/ims90-6.ps.Z . Published in  
1991 SIGGRAPH Course Notes #14: Fractal Modeling in 3D Computer  
Graphics and Imaging.  
  
Instead of quaternions, you can of course use other functions. For instance,  
you could use a map with more than one parameter, which would generate  
a higher-dimensional fractal.  
  
Another way of generating 3-D fractals is to use 3-D iterated function  
systems (IFS). These are analogous to 2-D IFS, except they generate points  
in a 3-D space.  
  
A third way of generating 3-D fractals is to take a 2-D fractal such as the  
Mandelbrot set, and convert the pixel values to heights to generate a 3-D  
"Mandelbrot mountain". This 3-D object can then be rendered with normal  
computer graphics techniques.  
  
------------------------------  
  
Subject: Fractint  
  
Q21a: What is Fractint?  
A21a: Fractint is a very popular freeware (not public domain) fractal  
generator. There are DOS, Windows, OS/2, and Unix/X versions. The  
DOS version is the original version, and is the most up-to-date. There is a  
new Amiga version.  
  
Please note: sci.fractals is not a product support newsgroup for Fractint.  
Bugs in Fractint/Xfractint should usually go to the authors rather than being  
posted.  
  
Fractint is on many ftp sites. For example:  
DOS: ftp from wuarchive.wustl.edu:/systems/ibmpc/simtel/graphics  
 [128.252.135.4]. The source is in the file frasr182.zip. The executable  
 is in the file frain182.zip. (The suffix 182 will change as new versions  
 are released.) Fractint is available on Compuserve: GO GRAPHDEV and look  
 for FRAINT.EXE and FRASRC.EXE in LIB 4.  
There is a collection of map, parameter, etc. files for Fractint, called  
 FracXtra. Ftp from wuarchive.wustl.edu:/systems/ibmpc/simtel/graphics.  
 File is fracxtr5.zip.  
Windows: ftp to wuarchive.wustl.edu:/systems/ibmpc/simtel/window3 .  
 The source is in the file wins1821.zip. The executable is in the file  
 winf1821.zip.  
OS/2: available on Compuserve in its GRAPHDEV forum. The files are  
PM*.ZIP.  
 These files are also available by ftp:  
 ftp-os2.nmsu.edu:/pub/os2/2.0/graphics in pmfra2.zip.  
Unix: ftp to sprite.berkeley.edu [128.32.150.27]. The source is in the file  
 xfract203.shar.Z. Note: sprite is an unreliable machine; if you can't  
 connect to it, try again in a few hours, or try hijack.berkeley.edu.  
 Xfractint is also available in LIB 4 of Compuserve's GO GRAPHDEV  
 forum in XFRACT.ZIP.  
Macintosh: there is no Macintosh version of Fractint, although there are  
 several people working on a port. It is possible to run Fractint on the  
 Macintosh if you use Insignia Software's SoftAT, which is a PC AT  
 emulator.  
Amiga: There is an Amiga version at 
wuarchive.wustl.edu:/pub/aminet/gfx/fract .  
  
For European users, these files are available from ftp.uni-koeln.de. If you  
can't use ftp, see the mail server information below.  
  
Q21b: How does Fractint achieve its speed?  
A21b: Fractint's speed (such as it is) is due to a combination of:  
  
1. Using fixed point math rather than floating point where possible (huge  
improvement for non-coprocessor machine, small for 486's).  
  
2. Exploiting symmetry of the fractal.  
  
3. Detecting nearly repeating orbits, avoid useless iteration (e.g. repeatedly  
iterating 0^2+0 etc. etc.).  
  
4. Reducing computation by guessing solid areas (especially the "lake"  
 area).  
  
5. Using hand-coded assembler in many places.  
  
6. Obtaining both sin and cos from one 387 math coprocessor instruction.  
  
7. Using good direct memory graphics writing in 256-color modes.  
  
The first four are probably the most important. Some of these introduce  
errors, usually quite acceptable.  
  
------------------------------  
  
Subject: Fractal software  
  
Q22a: Where can I obtain software packages to generate fractals?  
A22a:  
For X windows:  
 xmntns and xlmntn: these generate fractal mountains. They can be obtained  
  from ftp: ftp.uu.net:/usenet/comp.sources.x/volume8/xmntns  
  [137.39.1.9].  
 xfroot: generates a fractal root window.  
 xmartin: generates a Martin hopalong root window.  
 xmandel: generates Mandelbrot/Julia sets.  
 xfroot, xmartin, xmandel are part of the X11 distribution.  
 lyap: generates Lyapunov exponent images. Ftp from:  
  ftp.uu.net:/usenet/comp.sources.x/volume17/lyapunov-xlib .  
 spider: Uses Thurston's algorithm for computing postcritically finite  
  polynomials, draws Mandelbrot and Julia sets using the Koebe  
  algorithm, and draws Julia set external angles. Ftp from:  
  lyapunov.ucsd.edu:pub/inls-ucsd/spider .  
 xfractal: fractal drawing program. Ftp from: clio.rz.uni-  
  duesseldorf.de:/X11/uploads [134.99.128.3].  
  
Distributed X systems:  
 MandelSpawn: computes Mandelbrot/Julia sets on a network of  
 machines. Ftp  
  from: export.lcs.mit.edu:/contrib [18.24.0.12] or  
  funic.funet.fi:/pub/X11/contrib [128.214.6.100] in mandelspawn-  
  0.06.tar.Z.  
 gnumandel: computes Mandelbrot images on a network. Ftp from:  
  informatik.tu-muenchen.de:/pub/GNU/gnumandel [131.159.0.110].  
  
For SunView:  
 Mandtool: A Mandelbrot computing program. Ftp from:  
  spanky.triumf.ca:/fractals/programs/mandtool ; code is in M_TAR.Z .  
  
For Unix/C:  
 lsys: generates L-systems as PostScript or other textual output. No  
  graphical interface at present. (in C++) Ftp from:  
  ftp.cs.unc.edu:/pub/leech/lsys.tar.Z .  
 lyapunov: generates PGM Lyapunov exponent images. Ftp from:  
  ftp.uu.net:/usenet/comp.sources.misc/volume23/lyapuov . SPD: contains  
  generators for fractal mountain, tree, recursive tetrahedron. Ftp  
  from: princeton.edu:/pub/Graphics [128.112.128.1].  
 Fractal Studio: Mandelbrot set program; handles distributed computing.  
  Ftp from archive.cs.umbc.edu:/pub/peter/fractal-studio  
  [130.85.100.53].  
 Xmountains: An X11-based fractal landscape generator. Ftp from  
  ftp.epcc.ed.ac.uk:/pub/personal/spb/xmountains .  
  
For Mac:  
 LSystem, 3D-L-System, IFS, FracHill, Mandella and a bunch of others are  
  available from uceng.uc.edu:/pub/wuarchive/edu/math/mac/fractals  
  [129.137.189.1] or wuarchive.wustl.edu:/edu/math/mac/fractals .  
  (These are also available in New Zealand at ccu1.auckland.ac.nz.)  
 fractal-wizard.hqx, julias-dream-107.hqx, mandella-87.hqx, and others are  
  under app in the info-mac archive: sumex-aim.stanford.edu:/info-mac  
  [36.44.0.6], or a mirror such as  
  plaza.aarnet.edu.au:/micros/mac/info-mac [139.130.4.6].  
 mandel-tv: a very fast Mandelbrot generator. Under sci at info-mac.  
 There are also commercial programs, such as IFS Explorer and Fractal Clip  
 Art, which are published by Koyn Software (314) 878-9125.  
  
For NeXT:  
 Lyapunov: generates Lyapunov exponent images. Ftp from:  
  nova.cc.purdue.edu:/pub/next/2.0-release/source .  
  
For MSDOS:  
 DEEPZOOM: a high-precision Mandelbrot program for displaying highly zoomed  
  fractals. Ftp from spanky.triumf.ca [142.90.112.1] in  
  [pub.fractals.programs.ibmpc] depzm13.zip.  
 Fractal WitchCraft: a very fast fractal design program. Ftp from:  
  garbo.uwasa.fi:/pc/demo/fw1-08.zip [128.214.87.1].  
 CAL: generates more than 15 types of fractals including Mandelbrot,  
  Lyapunov, IFS, user-defined formulas, logistic equation, and  
  quaternion julia sets. Ftp from: oak.oakland.edu:/pub/msdos/graphics  
  [141.210.10.117] (or any other Simtel mirror) in frcal035.zip.  
 Fractal Discovery Laboratory: designed for use in a science museum or  
  school setting. The Lab has five sections: Art Gallery ( 72 images --  
  Mandelbrots, Julias, Lyapunovs), Microscope ( 85 images -- Biomorph,  
  Mandelbrot, Lyapunov, ...), Movies (165 images, 6 "movies":  
  Mandelbrot Evolution, Splitting a Mini-Mandelbrot, Fractal UFO, ...),  
  Tools (Gingerbreadman, Lorentz Equations, Fractal Ferns, von Koch  
  Snowflake, Sierpinski Gasket), and Library (Dictionary, Books and  
  Articles). Sampler available from Compuserver GRAPHDEV Lib 4 in  
  DISCOV.ZIP, or send high-density disk and self-addressed, stamped  
  envelope to: Earl F. Glynn, 10808 West 105th Street, Overland Park,  
  Kansas 66214-3057.  
 WL-Plot: plots functions including bifurcations and recursive relations.  
  Ftp from wuarchive.wustl.edu:/edu/math/msdos/graphing in wlplt231.zip.  
 There are many fractal programs available from  
  oak.oakland.edu:/pub/msdos/graphics [141.210.10.117]:  
  forb01a.zip: Displays orbits of Mandelbrot mapping. C/E/VGA  
   fract30.arc: Mandelbrot/Julia set 2D/3D EGA/VGA Fractal Gen  
   fractfly.zip: Create Fractal flythroughs with FRACTINT  
   fdesi313.zip: Program to visually design IFS fractals  
   frain182.zip: FRACTINT v18.1 EGA/VGA/XGA fractal generator  
   frasr182.zip: C & ASM src for FRACTINT v18.1 fractal gen.  
   frcal040.zip: Fractal drawing program: 15 formulae available  
   frcaldmo.zip: 800x600x256 demo images for FRCAL030.ZIP  
  
For Windows:  
 dy-syst.zip. This program explores Newton's method, Mandelbrot set, and  
  Julia sets. Ftp from mathcs.emory.edu:/pub/riddle .  
  
For Amiga: (all entries marked "ff###" are .lzh files in the Fish Disk set  
 available at ux1.cso.uiuc.edu:/amiga/fish and other sites)  
 General Mandelbrot generators with many features: Mandelbrot (ff030),  
  Mandel (ff218), Mandelbrot (ff239), TurboMandel (ff302), MandelBltiz  
  (ff387), SMan (ff447), MandelMountains (ff383, in 3-D), MandelPAUG  
  (ff452, MandFXP movies), MandAnim (ff461, anims), ApfelKiste (ff566,  
  very fast), MandelSquare (ff588, anims)  
 Mandelbrot and Julia sets generators: MandelVroom (ff215), Fractals  
  (ff371, also Newton-R and other sets)  
 With different algorithmic approaches (shown): FastGro (ff188, DLA),  
  IceFrac (ff303, DLA), DEM (ff303, DEM), CPM (ff303, CPM in 3-D),  
  FractalLab (ff391, any equation)  
 Iterated Function System generators (make ferns, etc): FracGen (ff188,  
  uses "seeds"), FCS (ff465), IFSgen (ff554), IFSLab (ff696, "Collage  
  Theorem")  
 Unique fractal types: Cloud (ff216, cloud surfaces), Fractal (ff052,  
  terrain), IMandelVroom (strange attractor contours?), Landscape  
  (ff554, scenery), Scenery (ff155, scenery), Plasma (ff573, plasma  
  clouds)  
 Fractal generators: PolyFractals (ff015), FFEX (ff549)  
 Lyapunov fractals: Ftp from: ftp.luth.se:/pub/aminet/new/lyapunovia.lha  
  [130.240.18.2].  
 Commercial packages: Fractal Pro 5.0, Scenery Animator 2.0, Vista  
  Professional, Fractuality (reviewed in April '93 Amiga User  
  International).  
 MathVISION 2.4. Generates Julia, Mandelbrot, and others. Includes  
  software for image processing, complex arithmetic, data display,  
  general equation evaluation. Available for $223 from Seven Seas  
  Software, Box 1451, Port Townsend WA 98368.  
  
Software for computing fractal dimension:  
 Fractal Dimension Calculator is a Macintosh program which uses the box-  
  counting method to compute the fractal dimension of planar graphical  
  objects. Ftp from:  
  wuarchive.wustl.edu:/edu/math/mac/fractals/FDC.sea.hqx or  
  wuarchive.wustl.edu:/packages/architec/Fractals/FDC.sea.hqx .  
 FD3: estimates capacity, information, and correlation dimension from a  
  list of points. It computes log cell sizes, counts, log counts, log  
  of Shannon statistics based on counts, log of correlations based on  
  counts, two-point estimates of the dimensions at all scales examined,  
  and over-all least-square estimates of the dimensions. Ftp from:  
  lyapunov.ucsd.edu:/pub/cal-state-stan [132.239.86.10]. Also look in  
  lyapunov.ucsd.edu:/pub/inls-ucsd for an enhanced Grassberger-Procaccia  
  algorithm for correlation dimension. A MS-DOS version of FP3 is  
  available by request to gentry@altair.csustan.edu.  
  
Q22b: What are some supporting software/utilities?  
A22b: Some supporting software/utilities/sources are:  
  
http://akebono.stanford.edu/yahoo/Computers/Software/Graphics/ Yahoo  
at Stanford University  
  
http://garnet.acns.fsu.edu/~swingree/eimaging.html Electronic Imaging 
Software   
  
http://www2.ncsu.edu/bae/people/faculty/walker/hotlist/graphics.html   
Graphics viewers, editors, utilities and info   
  
file://ftp.switch.ch/mirror/msdos/zip PKzip (pkz204g)   
  
file://ftp.switch.ch/mirror/msdos/windows3 WinZip   
  
ftp://ftp.cadence.com/pictures/index.html Compression Utilities   
  
file://gatekeeper.dec.com/.f/micro/msdos/win3/desktop/ima.zip Image'n Bits  
  
http://www.cis.ohio-state.edu/hypertext/FAQ/usenet/jpeg-FAQ/FAQ.html  
JPEG FAQ   
  
file://gatekeeper.dec.com/.f/micro/msdos/win3/desktop/lview31.zip Lview  
  
http://www.cm.cf.ac.uk:80/Ray.Tracing/ Ray Tracing   
  
ftp://oak.oakland.edu/pub/msdos/visbasic   
VBRUN (vbrun100.zip, vbrun200,zip, vbrun300.zip)   
  
file://gatekeeper.dec.com/.f/micro/msdos/win3/desktop/wingif14.zip WinGIF  
  
file://gatekeeper.dec.com/.f/micro/msdos/win3/desktop/winjp265.zip WinJPEG  
  
http://hoohoo.ncsa.uiuc.edu/archie.html Archie Search   
  
http://www.fagg.uni-lj.si/cgi-bin/shase Shareware Search Engine   
  
------------------------------  
  
Subject: Ftp questions  
  
Q23a: How does anonymous ftp work?  
A23a: Anonymous ftp is a method of making files available to anyone on  
the Internet. In brief, if you are on a system with ftp (e.g. Unix), you 
type "ftp lyapunov.ucsd.edu", or whatever system you wish to access. You 
are prompted for your name and you reply "anonymous". You are prompted  
for your password and you reply with your email address. You then use  
ls" to list the files, "cd" to change directories, "get" to get files, an
"quit" to exit. For example, you could say "cd /pub", "ls", "get README", 
and "quit"; this would get you the file "README". See the man page ftp(1) 
or ask someone at your site for more information.  
  
In this FAQ file, anonymous ftp addresses are given in the form  
name.of.machine:/pub/path [1.2.3.4]. The first part "name.of.machine" is  
the machine you must ftp to. If your machine cannot determine the host  
from the name, you can try the numeric Internet address: "ftp 1.2.3.4". The  
part after the colon: "/pub/path" is the file or directory to access once you  
are connected to the remote machine.  
  
Q23b: What if I can't use ftp to access files?  
A23b: If you don't have access to ftp because you are on a uucp/Fidonet/etc  
network there is an e-mail gateway at ftpmail@decwrl.dec.com that can  
retrieve the files for you. To get instructions on how to use the ftp gateway  
send a message to ftpmail@decwrl.dec.com with one line containing the  
word 'help'.  
  
------------------------------  
  
Subject: Archived pictures  
  
Q24a: Where are fractal pictures archived?  
A24a: Fractal images (GIFs, etc.) used to be posted to alt.fractals.pictures;  
this newsgroup has been replaced by alt.binaries.pictures.fractals. Pictures  
from 1990 and 1991 are available via anonymous ftp:  
csus.edu:/pub/alt.fractals.pictures [130.86.90.1].  
  
Many Mandelbrot set images are available via anonymous ftp:  
ftp.ira.uka.de/pub/graphic/fractals [129.13.10.93].  
  
Fractal images including some recent alt.binaries.pictures.fractals images are  
archived at spanky.triumf.ca:/fractals [142.90.112.1]. This can also be  
accessed via WWW at http://spanky.triumf.ca/ .  
  
Some fractal images are available on the WWW at  
http://www.cnam.fr/fractals.html . These images are available by ftp:  
ftp.cnam.fr:/pub/Fractals . Fractal animations in MPG and FLI format are  
in ftp.cnam.fr:/pub/Fractals/anim or http://www.cnam.fr/fractals/anim.html .  
Another collection of fractal images is archived at  
ftp.maths.tcd.ie/pub/images/Computer [134.226.81.10]. Some fractal and  
other computer-generated images are available on the WWW at  
gopher://olt.et.tudelft.nl:1251/11/computer .  
  
A collection of interesting smoke- and flame-like jpeg iterated function  
system images is available on the WWW at  
http://www.cs.cmu.edu:8001/afs/cs.cmu.edu/user/spot/web/images.html .  
Some images are also available from:  
ftp://hopeless.mess.cs.cmu.edu:/usr/spot/pub/film  
  
An algorithmic image gallery is available on the WWW at  
http://axpba1.ba.infn.it:8080/ .  
  
Other tutorials, resources, and galleries of images are:  
  
http://sprott.physics.wisc.edu/fractals.htm Fractal Gallery (J. C. Sprott)  
http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html Fractal Microscope   
  
http://is.dal.ca:3400/~adiggins/fractal/ Dalhousie University Fractal Gallery   
  
http://acat.anu.edu.au/contours.html "Contours of the Mind"   
  
http://www.maths.tcd.ie/pub/images/images.html Computer Graphics Gallery  
  
http://wwfs.aist-nara.ac.jp/shika/library/fractal/ SHiKA Fractal Image
Library  
  
http://www.awa.com/sfff/sfff.html The San Francisco Fractal Factory.   
  
http://spanky.triumf.ca/www/spanky.html Spanky (Noel Giffin)  
  
http://www.cnam.fr/fractals.html Fractal Gallery (Frank Rousell)   
  
http://www.cnam.fr/fractals/anim.html Fractal Animations Gallery  
(Frank Rousell)   
  
http://akebono.stanford.edu/yahoo/Art/Computer_Generated/Fractals/   
Stanford University Pointers   
  
http://axpba1.ba.infn.it:8080/ The Algorithmic Image Gallery  
(Giuseppe Zito)   
  
http://acat.anu.edu.au/works/gallery.html ANU Images   
  
http://www.geom.umn.edu/pix/archive/subjects/fractals.html   
Geometry Centre at University of Minnesota   
  
http://www.rain.org:80/~ayb/ Fractal Images (Art Baker) .  
  
http://acacia.ens.fr:8080/home/massimin/quat/quat.ang.html   
Quaternion Julia Set (Pascal Massimino)   
  
http://www.wri.com/~mathart/portfolio/SPD_Frac_portfolio.html   
3d Fractals (Stewart Dickson) via Mathart.com.  
  
http://irc.umbc.edu/gallery/Fractals/grindex.html Fractal Gallery   
  
http://sashimi.wwa.com:80/mirror/gallerie/fracgall/fg941101.htm   
volume fg941101 (Alan Beck-Virtual Mirror)   
  
http://www.softsource.com/softsource/fractal.html Softsource .  
  
http://www.ncsa.uiuc.edu/SDG/People/rgrant/fav_pics.html   
Favourite Fractals (Ryan Grant)   
  
ftp://csus.edu/pub/alt.fractals.pictures A.F.P. Fractal FTP Archive   
  
http://hydra.cs.utwente.nl/~schol/video.html Eric Schol   
  
http://aleph0.clarku.edu/~djoyce/home.html Mandelbrot and Julia Sets   
(David E. Joyce)   
  
http://aleph0.clarku.edu/~djoyce/newton/newton.html Newton's method .  
  
http://www.vanderbilt.edu/VUCC/Misc/Art1/fractals.html   
Gratuitous Fractals (evans@ctrvax.vanderbilt.edu)   
  
http://www.ccsf.caltech.edu/ismap/image.html Xmorphia   
  
  
Q24b: How do I view fractal pictures from alt.binaries.pictures.fractals?  
A24b: A detailed explanation is given in the "alt.binaries.pictures FAQ"  
(see "pictures-FAQ"). This is posted to the pictures newsgroups and is  
available by ftp: rtfm.mit.edu:/pub/usenet/news.answers/pictures-FAQ  
[18.181.0.24].  
  
In brief, there is a series of things you have to do before viewing these  
posted images. It will depend a little on the system your working with, but  
there is much in common. Some newsreaders have features to automatically  
extract and decode images ready to display ("e" in trn) but if you don't you  
can use the following manual method:  
  
1. Save/append all posted parts sequentially to one file.  
  
2. Edit this file and delete all text segments except what is between the  
BEGIN-CUT and END-CUT portions. This means that BEGIN-CUT and  
END-CUT lines will disappear as well. There will be a section to remove  
for each file segment as well as the final END-CUT line. What is left in the  
file after editing will be bizarre garbage starting with begin 660  
imagename.GIF and then about 6000 lines all starting with the letter "M"  
followed by a final "end" line. This is called a uuencoded file.  
  
3. You must uudecode the uuencoded file. There should be an appropriate  
utility at your site; "uudecode filename" should work under Unix. Ask a  
system person or knowledgeable programming type. It will decode the file and  
produce another file called imagename.GIF. This is the image file.  
  
4. You must use another utility to view these GIF images. It must be  
capable of displaying color graphic images in GIF format. (If you get a JPG  
format file, you may have to convert it to a GIF file with yet another 
utility.)  In the XWindows environment, you may be able to use "xv", 
"xview", or "xloadimage" to view GIF files. If you aren't using X, then 
you'll either have to find a comparable utility for your system or transfer 
your file to some other system. You can use a file transfer utility such 
as Kermit to transfer the binary file to an IBM-PC.  
  
An online resource that may be helpful is:  
  
ftp://ftp.cadence.com/pictures/index.html alt.binaries.pictures utilities  
archive   
  
------------------------------  
  
Subject: Where can I obtain fractal papers?  
  
Q25: Where can I obtain fractal papers?  
A25: There are several Internet sites with fractal papers:  
  
There is an ftp archive site for preprints and programs on nonlinear  
dynamics and related subjects at: lyapunov.ucsd.edu:/pub [132.239.86.10].   
There are also articles on dynamics, including the IMS preprint series,  
available from math.sunysb.edu:/preprints [129.49.31.57].  
  
A collection of short papers on fractal formulas, drawing methods, and  
transforms is available by ftp: ftp.coe.montana.edu:/pub/fractals (this site  
hasn't been working lately).  
  
The WWW site http://inls.ucsd.edu/y/complex.html has  
some fractal papers; they are also available by  
ftp://legendre.ucsd.edu:/pub/Research/Fisher .  
  
The site life.anu.edu.au [150.203.38.74] has a collection of fractal  
programs, papers, information related to complex systems, and gopher and  
World Wide Web connections. The ftp path is:
life.anu.edu.au:/pub/complex_systems .  Look in fractals, tutorial, and 
anu92. The Word Wide Web access is:  
http://life.anu.edu.au/complex_systems/complex.html. The gopher path is:  
Name=BioInformatics gopher at ANU  
Host=life.anu.edu.au  
Type=1  
Port=70  
Path=1/complex_systems/fractals  
  
------------------------------  
  
Subject: How can I join the BITNET fractal discussion?  
  
Q26: How can I join the BITNET fractal discussion?  
A26: There is a fractal discussion on BITNET that uses an automated mail  
server that sends mail to a distribution list. (On some systems, the contents  
of FRAC-L appear in the Usenet newsgroup bit.listserv.frac-l.) To join the  
mailing list, send a message to listserv@gitvm1.gatech.edu or  
listserv@GITVM1 with the following line of text:  
SUBSCRIBE FRAC-L John Doe  
(where John Doe is replaced by your name)  
To unsubscribe, send the message:  
UNSUBSCRIBE FRAC-L or SIGNOFF FRAC-L (GLOBAL)  
  
Messages posted to frac-l are archived along with several files. The index  
of the archive may be obtained by sending email to:  
listserv@GITVM1.BITNET or listserv@GITVM1.GATECH.EDU  
with the sole line of text in the body: INDEX FRAC-L  
  
Files identified in the index (filelist) may then be retrieved by sending  
another message to the listserv with the line of text: GET filename  
(where filename is replaced by the exact name of a file given in the index).  
  
If there is any difficulty contact the listowner: Ermel Stepp  
(stepp@marshall.edu.  
  
------------------------------  
  
Subject: Complexity  
  
Q27: What is complexity?  
A27: Emerging paradigms of thought encompassing fractals, chaos,  
nonlinear science, dynamic systems, self-organization, artificial life, 
neural networks, and similar systems comprise the science of complexity. 
Several helpful online resources on complexity are:  
  
http://www.marshall.edu/~stepp/vri/irc/irc.html  
Institute for Research on Complexity  
  
The site life.anu.edu.au [150.203.38.74] has a collection of fractal  
programs, papers, information related to complex systems, and gopher and  
World Wide Web connections.  
  
 The ftp path is life.anu.edu.au:/pub/complex_systems ; (look in  
 fractals, tutorial, and anu92).  
  
 The gopher path is:  
 gopher://life.anu.edu.au:70/1/complex_systems/fractals  
  
 The Word Wide Web access is  
 http://life.anu.edu.au/complex_systems/complex.html.  
  
http://www.seas.upenn.edu/~ale/cplxsys.html Complex Systems  
(UPENN)  
  
http://jaguar.cssr.uiuc.edu/CCSRHome.html Complex Systems Research  
(UIUC)  
  
http://life.anu.edu.au/ci/ci,html Complexity International Journal or  
ftp://life.anu.edu.au/pub/complex_systems/ci  
  
ftp://xyz.lanl.gov/nlin-sys Nonlinear Science Preprints  
  
Nonlinear Science Preprints via emaiL:  
To subscribe to public bulletin board to receive announcements of the  
availability of preprints from Los Alamos National Laboratory, send email  
to nlin-sys@xyz.lanl.gov containing the sole line of text:  
subscribe your-real-name  
  
------------------------------  
  
Subject: References  
  
Q28a: What are some general references on fractals, chaos, and  
complexity?  
A28a: Some references are:  
  
M. Barnsley, _Fractals Everywhere_, Academic Press Inc., 1988. ISBN  
0-12-079062-9. This is an excellent text book on fractals. This is probably  
the best book for learning about the math underpinning fractals. It is also a  
good source for new fractal types.  
  
M. Barnsley and L. Anson, _The Fractal Transform_, Jones and  
Bartlett, April, 1993. ISBN 0-86720-218-1. This book is a sequel to  
_Fractals Everywhere_. Without assuming a great deal of technical knowledge,  
the authors explain the workings of the Fractal Transform (tm). The Fractal  
Transform is the compression tool for storing high-quality images in a  
minimal amount of space on a computer. Barnsley uses examples and  
algorithms to explain how to transform a stored pixel image into its fractal  
representation.  
  
R. Devaney and L. Keen, eds., _Chaos and Fractals: The Mathematics  
Behind the Computer Graphics_, American Mathematical Society,  
Providence, RI, 1989. This book contains detailed mathematical  
descriptions of chaos, the Mandelbrot set, etc.  
  
R. L. Devaney, _An Introduction to Chaotic Dynamical Systems_,  
Addison- Wesley, 1989. ISBN 0-201-13046-7. This book introduces  
many of the basic concepts of modern dynamical systems theory and leads  
the reader to the point of current research in several areas. It goes 
into great detail on the exact structure of the logistic equation and 
other 1-D maps.  The book is fairly mathematical using calculus and topology.  
  
R. L. Devaney, _Chaos, Fractals, and Dynamics_, Addison-Wesley,  
1990. ISBN 0-201-23288-X. This is a very readable book. It introduces  
chaos fractals and dynamics using a combination of hands-on computer  
experimentation and precalculus math. Numerous full-color and black and  
white images convey the beauty of these mathematical ideas.  
  
R. Devaney, _A First Course in Chaotic Dynamical Systems, Theory  
and Experiment_, Addison Wesley, 1992. A nice undergraduate  
introduction to chaos and fractals.  
  
A. K. Dewdney, (1989, February). Mathematical Recreations. _Scientific   
American_, pp. 108-111.  
  
G. A. Edgar, _Measure Topology and Fractal Geometry_, Springer-  
Verlag Inc., 1990. ISBN 0-387-97272-2. This book provides the math  
necessary for the study of fractal geometry. It includes the background  
material on metric topology and measure theory and also covers topological  
and fractal dimension, including the Hausdorff dimension.  
  
K. Falconer, _Fractal Geometry: Mathematical Foundations and  
Applications_, Wiley, New York, 1990.  
  
J. Feder, _Fractals_, Plenum Press, New York, 1988. This book is  
recommended as an introduction. It introduces fractals from geometrical  
ideas, covers a wide variety of topics, and covers things such as time series  
and R/S analysis that aren't usually considered.  
  
Y. Fisher (Ed), _Fractal Image Compression: Theory and Application_.  
Springer Verlag, 1995.  
  
J. Gleick, _Chaos: Making a New Science_, Penguin, New York, 1987.  
  
B. Hao, ed., _Chaos_, World Scientific, Singapore, 1984. This is an  
excellent collection of papers on chaos containing some of the most  
significant reports on chaos such as ``Deterministic Nonperiodic Flow'' by  
E.N.Lorenz.  
  
H. Jurgens, H. O Peitgen, & D. Saupe. (1990, August).   
The Language of Fractals. _Scientific American_, pp. 60-67.  
  
H. Jurgens, H. O. Peitgen, H.O., & D. Saupe. (1992). _Chaos and   
Fractals: New Frontiers of Science_. New York: Springer-Verlag.  
  
S. Levy, _Artificial life : the quest for a new creation_, Pantheon  
Books, New York, 1992. This book takes off where Gleick left off. It  
looks at many of the same people and what they are doing post-Gleick.  
  
B. Mandelbrot, _The Fractal Geometry of Nature_, W. H. FreeMan,  
New York. ISBN 0-7167-1186-9. In this book Mandelbrot attempts to  
show that reality is fractal-like. He also has pictures of many different  
fractals.  
  
H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_, Springer-  
Verlag, New York, 1986. ISBN 0-387-15851-0. This book has lots of  
nice pictures. There is also an appendix giving the coordinates and constants  
for the color plates and many of the other pictures.  
  
H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,  
Springer-Verlag, New York, 1988. ISBN 0-387-96608-0. This book  
contains many color and black and white photographs, high level math, and  
several pseudocoded algorithms.  
  
H. Peitgen, H. Juergens and D. Saupe, _Fractals for the Classroom_,  
Springer-Verlag, New York, 1992. These two volumes are aimed at  
advanced secondary school students (but are appropriate for others too),  
have lots of examples, explain the math well, and give BASIC programs.  
  
H. Peitgen, H. Juergens and D. Saupe, _Chaos and Fractals: New  
Frontiers of Science_, Springer-Verlag, New York, 1992.  
  
C. Pickover, _Computers, Pattern, Chaos, and Beauty: Graphics from  
an Unseen World_, St. Martin's Press, New York, 1990. This book  
contains a bunch of interesting explorations of different fractals.  
  
J. Pritchard, _The Chaos Cookbook: A Practical Programming Guide_,  
Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-0304-6. It contains  
type- in-and-go listings in BASIC and Pascal. It also eases you into 
some of the mathematics of fractals and chaos in the context of graphical  
experimentation. So it's more than just a type-and-see-pictures book, but  
rather a lab tutorial, especially good for those with a weak or rusty (or 
even nonexistent) calculus background.  
  
P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of  
Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good  
book on L-systems, which can be used to model plants in a very realistic  
fashion. The book contains many pictures.  
  
M. Schroeder, _Fractals, Chaos, and Power Laws: Minutes from an  
Infinite Paradise_, W. H. Freeman, New York, 1991. This book contains a  
clearly written explanation of fractal geometry with lots of puns and word  
play.  
  
J. Sprott, _Strange Attractors: Creating Patterns in Chaos_, M&T  
Books (subsidary of Henry Holt and Co.), New York. " ISBN 1-55851-  
298-5. This book describes a new method for generating beautiful fractal  
patterns by iterating simple maps and ordinary differential equations. It  
contains over 350 examples of such patterns, each producing a  
corresponding piece of fractal music. It also describes methods for  
visualizing objects in three and higher dimensions and explains how to  
produce 3-D stereoscopic images using the included red/blue glasses. The  
accompanying 3.5" IBM-PC disk contain source code in BASIC, C, C++,  
Visual BASIC for Windows, and QuickBASIC for Macintosh as well  
as a ready-to-run IBM-PC executable version of the program. Available for  
$39.95 + $3.00 shipping from M&T Books (1-800-628-9658).  
  
D. Stein, ed., _Proceedings of the Santa Fe Institute's Complex  
Systems Summer School_, Addison-Wesley, Redwood City, CA, 1988.   
See especially the first article by David Campbell: ``Introduction to  
nonlinear phenomena''.  
  
R. Stevens, _Fractal Programming in C_, M&T Publishing, 1989  
ISBN 1-55851-038-9. This is a good book for a beginner who wants to  
write a fractal program. Half the book is on fractal curves like the Hilbert  
curve and the von Koch snow flake. The other half covers the Mandelbrot,  
Julia, Newton, and IFS fractals.  
  
I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.  
Blackwell, New York, 1989.  
  
T. Wegner and M. Peterson, _Fractal Creations_, The Waite Group,  
1991. This is the book describing the Fractint program.  
  
http:wwwrefs.html Web references to Julia and Mandelbrot sets   
  
http://alephwww.cern.ch/~zito/chep94sl/sd.html   
Dynamical Systems (G. Zito)   
  
http://alephwww.cern.ch/~zito/chep94sl/chep94sl.html   
Scanning huge number of events (G. Zito)   
  
http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html   
The Who Is Who Handbook of Nonlinear Dynamics   
  
Q28b: What are some relevant journals?  
A28b: Some relevant journals are:  
  
"Chaos and Graphics" section in the quarterly journal _Computers and  
Graphics_. This contains recent work in fractals from the graphics  
perspective, and usually contains several exciting new ideas.  
  
"Mathematical Recreations" section by I. Stewart in _Scientific  
American_.  
  
_Fractal Report_. Reeves Telecommunication Labs. West Towan House,  
Porthtowan, TRURO, Cornwall TR4 8AX, U.K.  
  
_FRAC'Cetera_. This is a gazetteer of the world of fractals and related 
areas, supplied on IBM PC format HD disk. FRACTCetera is the home of FRUG -  
the Fractint User Group. For more information, contact:  
Jon Horner, Editor, FRAC'Cetera  
Le Mont Ardaine, Rue des Ardains, St. Peters  
Guernsey GY7 9EU  
Channel Islands, United Kingdom.  
Email: 100112,1700@compuserve.com  
  
_Fractals, An interdisciplinary Journal On The Complex Geometry of  
Nature_. This is a new journal published by World Scientific. B.B  
Mandelbrot is the Honorary Editor and T. Vicsek, M.F. Shlesinger, M.M  
Matsushita are the Managing Editors). The aim of this first international  
journal on fractals is to bring together the most recent developments in the  
research of fractals so that a fruitful interaction of the various approaches  
and scientific views on the complex spatial and temporal behavior could  
take place.  
  
------------------------------  
  
 Subject: Notices  
  
Q29: Are there any special notices?  
  
NOTICE (from Michael Peters):  
  
HOP - Fractals in Motion  
  
opens the door to a completely new world of fractals!  
  
Based on almost 30 new Hopalong type formulas and loads of incredible  
special effects, it produces an unlimited variety of images/animations  
quite unlike anything you have seen before.  
  
HOP features Fractint-like parameter files, GIF read/write,  
MAP palette editor, a screensaver for DOS, Windows, and OS/2, and more.  
Math coprocessor (386 and above) and SuperVGA required  
  
"HOP was originally based on HOPALONG, the Barry Martin creation which  
was popularized by A.K. Dewdney in one of his Scientific American  
articles. The HOP authors have taken Martin's idea well beyond his  
original concept, and developed it to such a degree that you need to keep  
reminding yourself of its modest beginnings. This program illustrates  
compellingly how a fundamentally simple idea can be extended, through the  
use of various graphics techniques, into something far removed from its  
humble origins. Don't let the simple name fool you - this is serious,  
robust, user friendly, IMAGINATIVE software !"  
(Jon Horner, editor, FRAC'cetera)  
  
$30 shareware  
Written by Michael Peters and Randy Scott  
  
HOP is usually contained in a self-extracting HOPZIP.EXE file.  
Places to download HOPZIP.EXE from:  
  
Compuserve GRAPHDEV forum, lib 4  
The Well under ibmpc/graphics  
slopoke.mlb.semi.harris.com  
ftp.uni-heidelberg.de (under /pub/msdos/graphics)  
spanky.triumf.ca [128.189.128.27] (under pub.fractals.programs.ibmpc)  
  
HOP WWW page: http://rever.nmsu.edu/~ras/hop  
  
HOP mailing list: write to hop-request@acca.nmsu.edu  
  
To subscribe to the HOP mailing list, simply send a message with the  
word "subscribe" in the Subject: field. For information, send a message  
with the word "INFO" in the Subject: field.  
  
One thing that I forgot to mention about HOP is that it is contained in   
the December issue of Jon Horner's FRAC'cetera magazine, and that   
FRAC'cetera subscribers can register HOP for $20 instead of $30.  
  
NOTICE from J. C. (Clint) Sprott (SPROTT@juno.physics.wisc.edu):  
  
The program, Chaos Data Analyzer, which I authored is a research and 
teaching tool containing 14 tests for detecting hidden determinism in a  
seemingly random time series of up to 16,382 points provided by the user in  
an ASCII data file. Sample data files are included for model chaotic  
systems. When chaos is found, calculations such as the probability  
distribution, power spectrum, Lyapunov exponent, and various measures of  
the fractal dimension enable you to determine properties of the system  
Underlying the behavior. The program can be used to make nonlinear  
predictions based on a novel technique involving singular value  
decomposition. The program is menu-driven, very easy to use, and even  
Contains an automatic mode in which all the tests are performed in succession  
and the results are provided on a one-page summary.  
  
Chaos Data Analyzer requires an IBM PC or compatible with at least 512K  
of memory. A math coprocessor is recommended (but not required) to  
Speed some of the calculations. The program is available on 5.25 or 3.5"  
disk and includes a 62-page User's Manual. Chaos Data Analyzer is peer-  
reviewed software published by Physics Academic Software, a cooperative  
Project of the American Institute of Physics, the American Physical Society,  
And the American Association of Physics Teachers.  
  
Chaos Data Analyzer and other related programs are available from The  
Academic Software Library, North Carolina State University, Box 8202,  
Raleigh, NC 27695-8202, Tel: (800) 955-TASL or (919) 515-7447 or  
Fax: (919) 515-2682. The price is $99.95. Add $3.50 for shipping in U.S.  
or $12.50 for foreign airmail. All TASL programs come with a 30-day,  
money-back guarantee.  
  
NOTICE from Noel Giffin (noel@erich.triumf.ca):  
  
Welcome to the Spanky Fractal Database   
  
This is a collection of fractal's and fractal related material for free  
distribution on the net. Most of the software was gathered from various  
ftp sites on the internet and it is generally freeware or shareware. Please  
abide by the guidelines set down in the individual packages. I would also  
like to make a disclaimer here. This page points to an enormous amount  
of information and no single person has the time to thoroughly check it  
all. I have tested software when I had the resources, and read through  
papers when I had the time, but other than certifying that it is related to  
fractals I can't assume any other responsibility.   
  
Enjoy and discover.   
  
	The correct URL for this site is:  
  
	http://spanky.triumf.ca/  
  
------------------------------  
  
Subject: Acknowledgements  
  
Q30: Who has contributed to the Fractal FAQ?  
A30:  
  
Participants in the Usenet group sci.fractals and the listserv forum frac-l  
have provided most of the content of Fractal FAQ. For their help with this  
FAQ, special thanks go to:  
  
Alex Antunes, Steve Bondeson, Erik Boman, Jacques Carette, John Corbit,  
Abhijit Deshmukh, Tony Dixon, Robert Drake, Detlev Droege, Gerald  
Edgar, Gordon Erlebacher, Yuval Fisher, Duncan Foster, David Fowler,  
Murray Frank, Jean-loup Gailly, Noel Giffin, Earl Glynn, Jon Horner, Lamont  
Granquist, Luis Hernandez- Ure:a, Jay Hill, Arto Hoikkala, Carl Hommel,  
Robert Hood, Oleg Ivanov, Simon Juden, J. Kai-Mikael, Leon Katz, Matt  
Kennel, Tal Kubo, Jon Leech, Brian Meloon, Tom Menten, Guy Metcalfe,  
Eugene Miya, Lori Moore, Robert Munafo, Miriam Nadel, Ron Nelson,  
Tom Parker, Dale Parson, Matt Perry, Cliff Pickover, Francois Pitt, Kevin  
Ring, Michael Rolenz, Tom Scavo, Jeffrey Shallit, Rollo Silver, J. C. Sprott,  
Ken Shirriff, Gerolf Starke, Bruce Stewart, Dwight Stolte, Tommy Vaske,  
Tim Wegner, Andrea Whitlock, Erick Wong, Wayne Young, and others.  
  
Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for  
collecting many of the chaos definitions.  
  
------------------------------  
  
Subject: Copyright  
  
Q31: Copyright?  
A31: Copyright (c) 1995 Ermel Stepp; 1994, 1993 Ken Shirriff  
  
The Fractal FAQ was created by Ken Shirriff and edited by him through  
September 26, 1994. The current editor of the Fractal FAQ is Ermel Stepp.  
Standing permission is given for non-profit reproduction and distribution of  
this issue of the Fractal FAQ as a complete document. Contact the editor for  
further information:  
  
Dr. Ermel Stepp  
Editor, Fractal FAQ  
Marshall University  
Huntington, WV 25755-2440  
(stepp@marshall.edu).