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Subject: Fractal Frequently Asked Questions and Answers

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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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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).