Newsgroups: bit.listserv.frac-l
Date:         Mon, 7 Sep 1992 12:29:50 PDT
From:         Ken Shirriff <shirriff@ALLSPICE.BERKELEY.EDU>
Subject:      Frequently Asked Questions

This is a frequently asked questions file for the Usenet newsgroup
alt.fractals.  I am sending it to FRAC-L as it may be of interest to the
readers there.  This file is posted on alt.fractals every two weeks.  Would
it be useful to send it to FRAC-L regularly also?

A copy of this FAQ is archived at various places such as:
pit-manager.mit.edu [18.72.1.58]:/pub/usenet/news.answers/fractal-faq and
ftp.uu.net [137.39.1.9 or 192.48.96.9]:/usenet/news.answers/fractal-faq.Z .

I am happy to receive more information to add to this file.  Also, if you
can correct mistakes you find, let me know.  Remember, this file depends
on your feedback and contributions.

Please send additions, comments, errors, etc. to Ken Shirriff
(shirriff@sprite.Berkeley.EDU).

The questions which are answered are:
Q1: What is fractint?
Q2: Where can I obtain other software packages to generate fractals?
Q3: What if I can't use FTP to access files?
Q4: Where is alt.fractals.pictures archived?
Q5: I want to learn about fractals.  What should I read first?
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?
Q8a: How does complex arithmetic work?
Q8b: How does quaternion arithmetic work?
Q9a: What is an iterated function system (IFS)?
Q9b: What is the state of fractal compression?
Q10: How are fractal mountains generated?
Q11: What are plasma clouds?
Q12a: Where are the popular Lyapunov fractals described?
Q12b: What are Lyapunov exponents?
Q12c: How can Lyapunov exponents be calculated?
Q13: What is the logistic equation?
Q14: What is chaos?
Q15: What is nonlinearity? What are nonlinear equations?
Q16: What is a fractal? What are some examples of fractals?
Q17: What is fractal dimension? How is it calculated?
Q18: What is a strange attractor?
Q19: How can I join the BITNET fractal discussion?
Q20: How can 3-D fractals be generated?
Q21: What are some general references on fractals and chaos?

You can search for the question you're interested in in "rn" or "trn"
using "g^Q11" (that's lower-case g, up-arrow, Q, and a number) where "11" is
the question you wish.  Or you may browse forward using <control-G> to
search for a Subject: line.

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

Questions and answers

Subject: Fractint

Q1: What is fractint?
A1: 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.
    The Unix version is still somewhat buggy.

    Please note: alt.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 to wuarchive.wustl.edu [128.252.135.4]
      The source is in the file /mirrors/msdos/graphics/frasr172.zip
      The executable is in the file /mirrors/msdos/graphics/frain172.zip
    Windows: ftp to wuarchive.wustl.edu
      The source is in the file /mirrors/msdos/windows3/winfr171.zip
      The executable is in the file /mirrors/msdos/windows3/winsr171.zip
    OS/2: available on Compuserve in its COMART forum in Library 12
      The files are PM*.ZIP.  These files are also available from
      ftp-os2.nmsu.edu in /pub/os2/pmfract.zoo, and from hobbes.nmsu.edu.
    Unix: ftp to sprite.berkeley.edu [128.32.150.27]
      The source is in the file xfract105.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.
    For European users, these files are available from ftp.uni-koeln.de
    If you can't use ftp, see the mail server info in Q3.

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

Subject: Other fractal software

Q2: Where can I obtain other software packages to generate fractals?
A2: For X windows:
	xmntns and xlmntn: these generate fractal mountains
	    They can be obtained from ftp.uu.net in the directory
	    /usenet/comp.sources.x/volume8/xmntns.
	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 in /usenet/comp.sources.x/volume16/lyap.
    Distributed X systems:
	MandelSpawn: computes Mandelbrot/Julia sets on a network of machines.
	    Ftp from: export.lcs.mit.edu:/contrib/mandelspawn-0.06.tar.Z
	    or funic.funet.fi:/pub/X11/contrib/mandelspawn-0.06.tar.Z
	gnumandel: computes Mandelbrot images on a network.
	    Ftp from: informatik.tu-muenchen.de [131.159.0.110] in
	    /pub/GNU/gnumandel.

    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 in pub/lsys.tar.Z.
	lyapunov: generates PGM Lyapunov exponent images.  Ftp from:
	    ftp.uu.net in /usenet/comp.sources.misc/volume23/lyapuov.
	SPD: contains generators for fractal mountain, tree, recursive
	    tetrahedron.  Ftp from: princeton.edu [128.112.128.1] in
	    /pub/Graphics.

    For Mac:
	fractal, L-System, 3DL-System, IFS, FracHill are available from
	    ftphost.aukuni.ac.nz [130.216.1.5] in the architec directory.
	fractal-wizard-15.hqx, julias-dream-107.hqx, mandel-julia-10.hqx,
	    mandel-zot-304.hqx, mandel-zot-304-extensions.hqx,
	    mandel-zot-lyapunov-ext.hqx, mandel-zot-304-docs.hqx,
	    mandelbrot-10.hqx are available from sumex.stanford.edu in
	    /info-map/app.
	mandel-tv: a very fast Mandelbrot generator.  Ftp from:
	    uhunix2.uhcc.hawaii.edu [128.171.44.7] in
	    /pub/mac/da/mandel-tv.hqx.
	Mandella: a fractal generator drawing over 60 fractals.  Ftp from:
	    sumex-aim.stanford.edu in /info-mac/demo/Mandella-606d6.hqx.

    For NeXT:
	Lyapunov: generates Lyapunov exponent images.  Ftp from:
	    nova.cc.purdue.edu in /pub/next/2.0-release/source.

    For MSDOS:
	Fractal WitchCraft: a very fast fractal design program.  Ftp from:
	    garbo.uwasa.fi in /pc/demo/fw1-08.zip.
	CAL: generates 15 types of fractals including Mandelbrot, Lyapunov,
	    IFS, and user-defined formulas.  Ftp from:
	    oak.oakland.edu (or any other Simtel mirror) in
	    pub/msdos/graphics/frcal030.zip.

    There is an archive site for preprints and programs on nonlinear dynamics
    and related subjects at lyapunov.ucsd.edu [132.239.86.10].
    There are also articles on dynamics, including the IMS preprint series,
    available from math.sunysb.edu [129.49.31.57].

    Please inform me of any other programs you know of.

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

Subject: Ftp mail server

Q3: What if I can't use FTP to access files?
A3: 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 blank message to ftpmail@decwrl.dec.com with one line
    containing the work 'help'.

   This is a sample message of how to retrieve xfractint from
   sprite.Berkeley.EDU:

   % mail ftpmail@decwrl.dec.com
   Subject:  <ignored>
   reply <yourname>@<yoursite>
   connect sprite.berkeley.edu anonymous
   dir
   binary
   uuencode    /* note: this command is optional and the default is btoa */
   get xfract105.shar.Z
   quit

   That would retrieve a directory of the archive, then xfract105.shar.Z.
   To receive xfract105.shar.Z, you must set the server to "binary" mode
   because the file is compressed.  Compressed files are then either sent
   out uuencoded or btoa'd.  So, you must obtain copies of the programs
   'uudecode' (or 'atob') and 'uncompress' to unpack the files that you
   will receive.  (Most Unix systems have uudecode and uncompress.)
   Ask your local computer guru for clarification on how to do this.

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

Subject: Archived pictures

Q4: Where is alt.fractals.pictures archived?
A4: Alt.fractals.pictures is the newsgroup for fractal images (GIFs, etc.).
    The pictures are available via anonymous FTP from csus.edu
    [130.86.90.1] in /pub/alt.fractals.pictures.

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

Subject: Learning about fractals

Q5: I want to learn about fractals.  What should I read first?
A5: There is a book list at the end.  _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.

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

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).

Q6b: How is the Mandelbrot set actually computed?
A6b: The basic algorithm is:
     For each 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 iterating 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 takes 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.

     Critical points are important because every attracting cycle for a
     polynomial or rational function attracts at least one critical point.
     [P. Fatou].  Thus, testing the critical point shows if there is any
     stable attractive cycle.

     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 diverges.
      Also, note that z1=c, so if |c|>2, the sequence diverges.

Q6e: How can I speed up Mandelbrot set generation?
A6e: Look at R. Rojas: "A tutorial on efficient computer graphic
     representations of the Mandelbrot set", Computers and Graphics, 1991,
     V15 N1 pp91-100.

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...  The behavior of the
     approximations suggests that the limit is between 1.66 and 1.71.
     However, the estimates of the area from below, using pixel counting,
     show that the area is at least 1.52.  The large gap between the lower
     bound 1.52 and the upper bound 1.71 may possibly be an indication that
     the boundary of the Mandelbrot set has positive area.
     Ref: J. H. Ewing and G. Schober, The area of the Mandelbrot set,
     Numer. Math. 61 (1992), 59-72.

Q6g: What can you say about the structure of the Mandelbrot set?
A6g: Just about all you could want to know is in Branner's article in
    _Chaos and Fractals: The Mathematics Behind the Computer Graphics_.

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 Mandlebrot 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.
     (Definitions 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.

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.

Q7c: How is a Julia set actually computed?
A7c: The Julia set can be computed by iteration similar to the Mandelbrot
     computation.
     Alternatively, points on the boundary of the Julia set can be computed
     quickly by using inverse iterations.

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

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)
     i^2 = -1
     exp(a+i*b) = exp(a)(cos(b)+i*sin(b))
     From these:
     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)
     Sine: sin(x) = (exp(i*x)-exp(-i*x))/(2*i)
     Cos: 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.)
     Complex powers: x^y = exp(y*log(x))
     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: Iterated function systems

Q9a: What is an iterated function system (IFS)?
A9a: If a fractal is self-similar, you can specify various mappings that map
     the whole onto the parts.  By taking a point and repeatedly applying
     these mappings you end up with a collection of points on the fractal.
     Iterated function systems can be used to make fractal ferns and trees
     and are also used in fractal image compression.
     _Fractals Everywhere_ by Barnsley is mostly about iterated function
     systems.

Q9b: What is the state of fractal compression?
A9b: According to Barnsley's book 'Fractals Everywhere', this method is
     based on a measure of deviation between a given image and its
     approximation by an IFS code.  The Collage Theorem states that there is
     a convergent process to minimize this deviation.  Unfortunately,
     according to an article Barnsley wrote for BYTE a few years ago, this
     convergence was rather slow, about 100 hours on a Cray, unless assisted
     by a person.

     Barnsley et al are not divulging any technical information beyond the
     meager bit in 'Fractals Everywhere'.  The book explains the idea of IFS
     codes at length, but is vague about the application of the Collage
     theorem to specific compression problems.

     There is reason to believe that Barnsley's company has
     *no algorithm* which takes a given reasonable image and achieves
     the compression ratios initially claimed for their fractal methods.
     The 1000-to-1 compression advertised was achieved only for a 'rigged'
     class of images, with human assistance. The best unaided
     performance I've heard of is good lossy compression of about 80-1.

     (This answer is from Tal Kubo <kubo@zariski.harvard.edu> in the
     comp.compression FAQ file.  There is a long list of references there.)

     A paper on the algorithm is supposed to come out this spring in
     IEEE Transactions on Signal Processing.  (More details are welcome.)

     Some sort of fractal compression demo program is available by
     anonymous ftp to lyapunov.ucsd.edu [132.239.86.10] in
     /pub/inls-ucsd/fractal-2.0.

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

Subject: Fractal mountains

Q10: How are fractal mountains generated?
A10: 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.

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

Subject: Plasma clouds

Q11: What are plasma clouds?
A11: 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

Q12a: Where are the popular Lyapunov fractals described?
A12a: Scientific American September 1991
     "Lyapunov Exponents of the Logistic Map with Periodic Forcing", M. Markus,
     B. Hess, Computers and Graphics, Vol 13, Number 4, pp553-558, 1989.

Q12b: What are Lyapunov exponents?
A12b: -A generalization of an eigenvalue.
      -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
       in one iteration.  Lambda is the Lyapunov exponent.  At least one
       Lyapunov exponent must be positive in a strange attractor.  A simple
       derivation is available in "Deterministic Chaos" by Heinz Georg
       Schuster, Physics Verlag, 1984, p. 18.

Q12c: How can Lyapunov exponents be calculated?
A12c: 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.

      Lyapunov Exponent References:

      1. Frank, M. and T. Stenges (1988) Journal of Economic Surveys 2,
         103-133.
      2. T.S. Parker and L.O. Chua (1989) "Practical Numerical Algorithms for
         Chaotic Systems." Springer Verlag, Ch. 3
      3. Scientific American, Sept. 1991

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

Subject: Logistic equation

Q13: What is the logistic equation?
A13: 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.
     For larger c, the population will oscillate between two values, then four
     values, eight, sixteen, etc.
     For still larger c (between 3.57 and 4), the population behavior is
     chaotic (for most c values).

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

Subject: Chaos

Q14: What is chaos?
A14: -An attractor is chaotic if at least one of its Lyapunov exponents
      is positive.  Chaos results from the existence of a chaotic attractor.
     -Chaos is the recurrent behavior of a deterministic dynamical system
      in which the phase-space divergence of nearby trajectories at an
      exponential rate results in a limited predictability horizon.
     -In chaotic iterated systems of the form x_{i+1}=f(x_i), the result
      after iteration is extremely sensitive to the initial value such that
      f^n(x_0+(epsilon)) is nowhere near f^n(x_0).
     -Chaos results from our inability to predict the future behavior of a
      deterministic system from initial conditions because of its great
      sensitivity to initial conditions.
     -Apparently unpredictable behavior arising in a deterministic system.
     -Unpredictable yet nonrandom behavior of a system.
     -Stochastic behavior occurring in a deterministic system.
      (Ian Stewart's _Does God Play Dice?_)

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

Subject: Nonlinearity

Q15: What is nonlinearity? What are nonlinear equations?
A15: -Nonlinear maps fail to satisfy the condition that
        f(ax+by)=af(x)+bf(y)
      where x and y are vectors, and a and b are scalars.
      e.g. f(x)=ax is linear.
           f(x)=x^2 is nonlinear.
      Nonlinearity is a map or term that is not linear.
      Nonlinear equations are equations containing nonlinearities.
      e.g. f(x)=ax+x^2 and f(x)=ax^2+bx+c are nonlinear.
     -A nonlinear system gives an output which is not proportional to the
      corresponding input.  Nonlinear dynamical systems possess nonlinear
      dynamical laws, which are functions of the system's state variables.
     -In linear systems, dy/dx is a constant, while in nonlinear systems
      dy/dx=some nonconstant function of x.
     -Nonlinear equations fail to exhibit linear superimposability.
      Nonlinear equations can be categorized by differentiability,
      discontinuity, and "memory" (e.g. hysteresis in an electric circuit),
      etc.  This can be important to some types of nonlinear analysis such
      as the Popov hyperstability criterion.

      Nonlinearity References:

      1. Zelinsky, D. "A First Course in Linear Algebra" (1973) p. 65. Academic
         Press.
      2. W.A. Brock and E.G. Baek "Some theory of statistical inference for
	 nonlinear science." (1991) Review of Economic Studies 58(4), 697-716.
      3. Guckenheimer, John and Philip Holmes.  _Nonlinear Oscillations,
	 Dynamical Systems, and Bifurcations of Vector Fields_.  Springer-
	 Verlag, New York, 1983.

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

Subject: What is a fractal?

Q16: What is a fractal? What are some examples of fractals?
A16: -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.  (A definition from B. Mandelbrot)
     -A set of points whose fractal dimension exceeds its topological
     dimension.
     -Examples of fractals:
       Sierpinski triangle, Koch snowflake, Peano curve, Mandlebrot set.

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

Subject: Fractal dimension

Q17: What is fractal dimension? How is it calculated?
A17: -Fractal dimension is the Hausdorff-Besikovich Dimension.
     -Roughly, fractal dimension can be calculated by taking the limit of the
      quotient of the log change in object size and the log change in
      measurement scale, as the measurement scale approaches zero.
      e.g.
      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.

      Fractal Dimension References:

      1. Barnsley et. al., "Science of Fractal Images", Springer Verlag, 1985.
      2. Falconer, K.J. (1985) "The Geometry of Fractal Sets" Cambridge Univ.
         Press.
      3. T.S. Parker and L.O. Chua (1989) "Practical Numerical Algorithms for
         Chaotic Systems."  Springer Verlag, Ch. 7.
      4. J.P. Eckmann and D. Ruelle (1985) Reviews of Modern Physics 57(3)
         617-656
      5. J. Theiler (1990) Physical Review A 41, 3038-3051.
      6. Grassberger and Procaccia (1983) Physica D 9, 189-208.

      References on how to estimate fractal dimension:
      1. "Estimating Fractal Dimension", J. Theiler, Journal of the Optical
         Society of America A-Optics and Image Science, June 1990
	 V7 N6 1055-1073.
      2. E. Peters, "Chaos and Order in the Capital Markets", John Wiley &
	 Sons, New York.  ISBN 0-471-53372-6 (1991).  Discusses methods
	 of computing fractal dimension.  Includes several short programs
	 for nonlinear analysis.

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

Subject: Strange attractors

Q18: What is a strange attractor?
A18: -An indecomposable closed invariant set that "attracts" the points about
      it which contains a transversal homoclinic orbit.  (This orbit accounts
      for the strangeness.)
     -The limit set of a chaotic trajectory.
     -A phase space locus of a bounded long-term dynamical behavior which has
      a nonzero probability of being observed - its basin of attraction has
      positive measure - and contains not a smooth manifold structure, but
      rather a self-similar or fractal structure.
      Note: While all chaotic attractors are strange, not all strange
	  attractors are chaotic. (Grebogi: et. al. (1984) "Strange Attractors
	  that are not chaotic." Physica D13, 261-268.)
     -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.
      Note: A strange attractor results if a system is sensitive to initial
            conditions and is not conservative.
     -The surfaces which the state of a chaotic system will be confined to,
      given time for transients to die out.


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

Subject: How can I join the BITNET fractal discussion?

Q19: How can I join the BITNET fractal discussion?
A19: There is a fractal discussion on BITNET that uses an automatic mail
     server that sends mail to a distribution list.  To join it, send
     a message to listserv@gitvm1.bitnet 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: 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:
     "Generation and Display of Geometric Fractals in 3-D", A. Norton,
     SIGGRAPH '82, pp61-67.
     "Ray Tracing Deterministic 3-D Fractals", J. Hart, D. Sandin, L.
     Kauffman, SIGGRAPH '89, pp289-296.

     Instead of quaternions, you can of course use other functions.  For
     instance, you could use the regular Mandelbrot formula, but vary both
     c and z0, resulting in a 4-D Mandelbrot set.

     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.

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

Subject: What are some general references?

Q21: What are some general references on fractals and chaos?
A21: Some references are:

_The Fractal Geometry of Nature_, Benoit Mandlebrot, New York: W. H.
    FreeMan and Company.  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.
_The Beauty of Fractals_, H. O. Peitgen and P. H. Richter, New York:
    Springer-Verlag Inc.  1986.  ISBN 0-387-15851-0.  Lots of neat pictures.
    There is also an appendix giving the coordinates and constants for the
    color plates and many of the other pictures.
_The Science of Fractal Images, ed.  H. O. Peitgen and P. H. Richter, New York:
    New York: Springer-Verlag Inc.  1988.  ISBN 0-387-96608-0.  This
    book contains many color and black and white photographs, high level math,
    and several psuedocoded algorithms.
_Fractals Everywhere_, Michael Barnsley, 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.
_Chaos_, James Gleick (1987)
_Does God Play Dice?_, Ian Stewart (1990)
_Chaos, Fractals, and Dynamics_, Robert L. Devaney, 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.
_Fractal Geometry_, Ken Falconer
_An Introduction to Chaotic Dynamical Systems_, Robert L. Devaney,
    Second Edition.  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.
Proceedings of the Santa Fe Institute's Complex Systems Summer School (1988)
    Official Title: Volume I "Lectures in the Sciences of Complexity"
    ed. Dan Stein.  See esp. the first article by David Campbell: "Introduction
    to nonlinear phenomena."
_Chaos and Fractals: The Mathematics Behind the Computer Graphics_, R.
    Devaney and L. Keen.  This book contains detailed mathematical
    descriptions of chaos, the Mandelbrot set, etc.
_Computers, Pattern, Chaos, and Beauty_, C. Pickover.  This book
    contains a bunch of interesting explorations of different fractals.
_Fractal Creations_.  This is the book describing the fractint program.
_Fractals for the Classroom_, Peitgen, Jurgens, Saupe.  This book
    is aimed at advanced secondary school students (but is appropriate
    for others too), has lots of examples, explains the math well, and
    gives BASIC programs.
_Fractal Programming in C_, Roger Stevens, 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
    Mandlebrot, Julia, Newton, and IFS fractals.
_Fractals, Chaos, and Power Laws_, M. Schroeder.  This book contains a
    clearly written explanation of fractal geometry, with lots of
    puns and word play.
_Fractals_, J. Feder.  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.
_Measure, Topology, and Fractal Geometry._, Gerald A. Edgar,  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.
_Chaos_, H. B. Lin.  This is an excellent collection of papers on chaos,
    containing some of the most significant reports on chaology, such as
    "deterministic nonperiodic flow" by E.N.Lorenz.

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 in Scientific American.
Algorithm
Mandala
Amygdala.  P.O. Box 219 San Cristobal, NM  87564-0219.  This is a
    newsletter about the Mandelbrot Set and other fractals.

Articles:
"Complex Analytic Dynamics on the Riemann Sphere", P. Blanchard, Bull. of
    the Amer. Math. Soc, 11(1), July 1984, p85-141.  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: Acknowledgements

For their help with this file, thanks go to:
Erik Boman, Abhijit Deshmukh, Robert Drake, Gerald Edgar, Gordon Erlebacher,
Duncan Foster, Murray Frank, Lamont Granquist, Arto Hoikkala, Carl Hommel,
Robert Hood, Oleg Ivanov, J. Kai-Mikael, Leon Katz, Matt Kennel, Tal Kubo,
Jon Leech, Tom Menten, Guy Metcalfe, Miriam Nadel, Ron Nelson, Tom Parker,
Matt Perry, Michael Rolenz, Tom Scavo, Jeffrey Shallit, Rollo Silver,
Bruce Stewart, Dwight Stolte, Tommy Vaske, Andrea Whitlock, Erick Wong,
and others.

Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for collecting
many of the chaos definitions.

Copyright 1992 Ken Shirriff (shirriff@sprite.Berkeley.EDU)
Comments: Gated by NETNEWS@AUVM.AMERICAN.EDU