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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  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 .  

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