Path: bloom-beacon.mit.edu!hookup!ames!agate!sprite.berkeley.edu!shirriff
From: shirriff@sprite.berkeley.edu (Ken Shirriff)
Newsgroups: sci.fractals,news.answers,sci.answers
Subject: Fractal Questions and Answers
Supersedes: <fractal-faq_766355035@sprite.Berkeley.EDU>
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Date: 8 May 1994 23:06:53 GMT
Organization: University of California, Berkeley
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Summary: Fractal software, algorithms, definitions, and references.
Keywords: fractals, chaos, Mandelbrot
Xref: bloom-beacon.mit.edu sci.fractals:3540 news.answers:19314 sci.answers:1138

Archive-name: fractal-faq
Last-modified: Mar 20, 1994

The international computer network Usenet contains discussions on a variety of
topics.  The Usenet newsgroup "sci.fractals" is devoted to discussions on
fractals.  Since many common questions reoccur during the discussions, I have
compiled this "Frequently Asked Questions" file, consisting of questions and
answers contributed by many participants.  This file also lists various pro-
grams and papers that can be accessed over the Internet by using "anonymous
ftp".  This file is not intended as a general introduction to fractals, or a
set of rigorous definitions, but rather a useful summary from sci.fractals.

* As a new feature, the fractal FAQ has some links for use with the World Wide
Web.  It can be accessed with a program such as xmosaic at
http://www.cis.ohio-state.edu/hypertext/faq/usenet/fractal-faq/faq.html .
Please let me know if there are more links I should add.

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 are some general references on fractals and chaos?

If you are viewing this file with a newsreaders such as "rn" or "trn", you can
search for a particular question by using "g^Q5" (that's lower-case g, up-
arrow, Q, and a number) where "5" is the question you wish.  Or you may browse
forward using <control-G> to search for a Subject: line.

This file is normally posted to the Usenet groups sci.fractals, news.answers,
and sci.answers about every two weeks.  Like most FAQs, the most recent copy
of this FAQ can be obtained over the Internet for free by "anonymous ftp" to
rtfm.mit.edu [18.70.0.209]; it is in /pub/usenet/news.answers/fractal-faq.

I am happy to receive more information to add to this file.  Also, let me know
if you find mistakes.  Please send additions, comments, errors, etc. to Ken
Shirriff (email: shirriff@cs.Berkeley.EDU, WWW:
file://sprite.berkeley.edu/www/ken.shirriff.html )

This file is Copyright 1993,1994 Ken Shirriff.  Permission is given for non-
profit distribution of this file, as long as the copyright notice and the list
of contributors remain attached.  However, I would like to be informed if you
distribute this file on other systems, so I have an idea of where it is.  Con-
tact me for more information on distribution.


------------------------------

Subject: Learning about fractals

Q1: I want to learn about fractals.  What should I read 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?").

------------------------------

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 di-
mension.  However, he is not satisfied with this definition as it excludes
sets one would consider fractals.

------------------------------

Subject: Chaos

Q3: What is chaos?
A3: Chaos is apparently unpredictable behavior arising in a deterministic sys-
tem 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.

------------------------------

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 ob-
ject.  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, correspond-
ing 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 .

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.

------------------------------

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 .

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
[18.70.0.209].

Two 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.

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 .

------------------------------

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 and a L-system leaf can
be viewed at gopher://life.anu.edu.au:70/I9/.WWW/complex_systems/leaf.gif .

------------------------------

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.

A IBM-PC program for fractal music is available by ftp to spanky.triumf.ca
[142.90.112.1] in [pub.fractals.programs.ibmpc] WTF23.ZIP.


------------------------------

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.

------------------------------

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.

------------------------------

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.

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?  *A: 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:/mirrors/msdos/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:/pub/MSDOS_UPLOADS/graphics.  File
    is fracxtr5.zip.
Windows: ftp to wuarchive.wustl.edu:/mirrors/msdos/window3 .  The source is in
    the file winsr1821.zip.  The executable is in the file winfr1821.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.

Q21a: How does Fractint achieve its speed?
A21a: 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

Q21b: Where can I obtain software packages to generate fractals?
A21b:
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/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].

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].
    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.  Obtain from hilljr@jupiter.saic.com .
    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:/pub/msdos_uploads/misc 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:/mirrors4/architec/Fractals/FracDim.sit.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.


------------------------------

Subject: Ftp questions

Q22: How does anonymous ftp work?
A22: 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, and "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.

Q23a: What if I can't use ftp to access files?
A23a: 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

Q23b: Where are fractal pictures archived?
A23b: 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/graphics/fractals [129.13.10.93].

Fractal images including some recent alt.binaries.pictures.fractals images are
archived at spanky.triumf.ca:/fractals [128.189.128.27].

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 by ftp: hopeless.mess.cs.cmu.edu:/usr/spot/pub/film
.

Q24a: How do I view fractal pictures from alt.binaries.pictures.fractals?
A24a: 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
[18.70.0.209].

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.

------------------------------

Subject: Where can I obtain fractal papers?

Q24b: Where can I obtain fractal papers?
A24b: 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 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

The WWW site http://legendre.ucsd.edu/Research/Fisher/complex.html has some
fractal papers; they are also available by ftp:
legendre.ucsd.edu:/pub/Research/Fisher .

One WWW site listing many other sites related to complex systems is
http://www.seas.upenn.edu/~ale/cplxsys.html .

------------------------------

Subject: How can I join the BITNET fractal discussion?

Q25: How can I join the BITNET fractal discussion?
A25: There is a fractal discussion on BITNET that uses an automatic 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.) Note that once
you join, you may have a very difficult time unsubscribing.  To join the
mailing list, send a message to listserv@gitvm1.gatech.edu with the following
as text:
SUBSCRIBE FRAC-L John Doe    (where John Doe is replaced by your name)
To unsubscribe, send the message:
UNSUBSCRIBE FRAC-L
If that doesn't unsubscribe you, you can try:
SIGNOFF FRAC-L (GLOBAL
If that doesn't work or you have other problems, you can contact the list
administrator.  You can obtain their name by sending the message:
REVIEW FRAC-L

------------------------------

Subject: References

Q26: What are some general references on fractals and chaos?
A26: Some references 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. 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.

3.  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.

4.  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.

5.  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.

6.  R. Devaney, _A First Course in Chaotic Dynamical Systems, Theory and
Experiment_, Addison Wesley, 1992.  A nice undergraduate introduction to chaos
and fractals.

7.  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.

8.  K. Falconer, _Fractal Geometry: Mathematical Foundations and
Applications_, Wiley, New York, 1990.

9.  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.

10.  J. Gleick, _Chaos: Making a New Science_, Penguin, New York, 1987.

11.  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.

12.  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.

13.  B. Mandelbrot, _The Fractal Geometry of Nature_, W. H.  FreeMan and Co.,
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.

14.  H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_, Springer-
Verlag Inc., 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.

15.  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.

16.  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.

17.  H. Peitgen, H. Juergens and D. Saupe, _Chaos and Fractals: New Frontiers
of Science_, Springer-Verlag, New York, 1992.

18.  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.

19.  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 non-
existent) calculus background.

20.  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.

21.  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.

22.  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).

23.  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''.

24.  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.

25.  I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
Blackwell, New York, 1989.

26.  T. Wegner and M. Peterson, _Fractal Creations_, The Waite Group, 1991.
This is the book describing the Fractint program.

Journals:

"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 A. K. Dewdney in _Scientific American_.

Algorithm - The Personal Computer Newsletter.  P.O. Box 29237, Westmount
Postal Outlet, 785 Wonderland Road S., London, Ontario, Canada, N6K 1M6.

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 in IBM PC format HD disk.  For more information, contact:  Jon
Horner, Editor, FRAC'Cetera, Le Mont Ardaine, Rue des Ardains, St. Peters,
Guernsey GY7 9EU, Channel Islands, United Kingdom.

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: Acknowledgements

For their help with this file, 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, 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, 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.

Copyright 1993,1994 Ken Shirriff (shirriff@cs.Berkeley.EDU).
