From: shirriff@sprite.berkeley.edu (Ken Shirriff)
Newsgroups: alt.fractals,news.answers
Subject: alt.fractals FAQ
Summary: Fractal software, algorithms, definitions, and references.
Keywords: fractals, chaos, Mandelbrot
Date: 15 Apr 92 19:34:32 GMT
Article-I.D.: sprite.faq_703366483
Expires: 9 May 1992 19:34:43 GMT
Followup-To: alt.fractals
Organization: University of California, Berkeley
Supersedes: <faq_702159964@sprite.Berkeley.EDU>
NNTP-Posting-Host: hijack.berkeley.edu

Archive-name: fractal-faq
Last-modified: April 8, 1992

This file is a frequently asked questions file for alt.fractals.
The purpose of this file is to collect common fractal questions and answers
into a convenient file.

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 include:

Q1: Where can I get a copy of fractint?
Q2: Where can I obtain other software packages to generate fractals of various
    types?
Q3: What if I can't use ftp?
Q4: Where is alt.fractals.pictures archived?
Q5: I want to learn about fractals.  What should I read first?
Q6: What is the Mandelbrot set?
Q7: What is the area of the Mandelbrot set?
Q8: How is the Mandelbrot set actually computed?
Q9: How can I speed up Mandelbrot set generation?
Q10: What is the difference between the Mandelbrot set and a Julia set?
Q11: What is the connection between the Mandelbrot set and Julia sets?
Q12: How is a Julia set actually computed?
Q13: How does complex arithmetic work?
Q14: What is an iterated function system?
Q15: How are fractal mountains generated?
Q16: What are plasma clouds?
Q17: Where are the popular Lyapunov fractals described?
Q18: What is the logistic equation?
Q19: What is chaos?
Q20: What is nonlinearity? What are nonlinear equations?
Q21: What is a fractal? What are some examples of fractals?
Q22: What is fractal dimension? How is it calculated?
Q23: What is a strange attractor?
Q24: What are Lyapunov exponents? How are they calculated?

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.  The Subject: lines and the lines of dashes
are an experiment; I am interested in knowing if anyone finds this format
better.

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

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 an alpha version with lots of bugs.

    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.
    Unix: ftp to sprite.berkeley.edu [128.32.150.27] (note: unreliable machine)
      The source is in the file xfract100.shar.Z
    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 of various
    types?
A2: For X windows:
	xmntn and xlmntn: these generate fractal mountains
	    They can be obtained from ftp.uu.net in the directory
	    /usenet/comp.sources.x/volume8/xmntn.
	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.
	    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.

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

    Please inform me of any other programs you know of.

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

Subject: Ftp mail server

Q3: What if I can't use FTP?
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
   uuencode    /* note: this command is optional and the default is btoa */
   binary
   get xfract100.shar.Z
   quit

   That would retrieve a directory of the archive, then xfract100.shar.Z.
   To receive xfract100.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: 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.
    _Chaos, Fractals, and Dynamics_ is also a good start.
    (Please send me suggestions on which books are best to start with.
    I'd also like brief descriptions of the books.)

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

Subject: Mandelbrot set

Q6: What is the Mandelbrot set?
A6: 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).

Q6a: What can you say about the structure of the Mandelbrot set?
A6a: Just about all you could want to know is in Branner's article in
    _Chaos and Fractals: The Mathematics Behind the Computer Graphics_.

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

Subject: Mandelbrot area

Q7: What is the area of the Mandelbrot set?
A7: 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.

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

Subject: Mandelbrot computation

Q8: How is the Mandelbrot set actually computed?
A8: 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.

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

Subject: Mandelbrot speedup

Q9: How can I speed up Mandelbrot set generation?
A9: Look at R. Rojas: "A tutorial on efficient computer graphic representations
    of the Mandelbrot set", Computers and Graphics, 1991, V15 N1 p91-100.

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

Subject: Julia set

Q10: What is the difference between the Mandelbrot set and a Julia set?
A10: 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.

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

Subject: Mandelbrot and Julia sets

Q11: What is the connection between the Mandelbrot set and Julia sets?
A11: 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.

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

Subject: Julia set computation

Q12: How is a Julia set actually computed?
A12: 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

Q13: How does complex arithmetic work?
A13: 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)
     Complex powers: x^y = exp(y*log(x))
     More details can be found in any complex analysis book.

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

Subject: Iterated function systems

Q14: What is an iterated function system?
A14: 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.

Q14a: What is the state of fractal compression?
A14a: 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.)

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

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.

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

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

Q17: Where are the popular Lyapunov fractals described?
A17: Scientific American September 1991, and Computers and Graphics, Vol 13,
     Number 4.

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

Subject: Logistic equation

Q18: What is the logistic equation?
A18: 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

Q19: What is chaos?
A19: -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?_)
  
     Chaos References:
     1. Devaney - An Introduction to Chaotic Dynamical Systems
	 (second edition)  Addison-Wesley:  Redwood City, CA, 1989
     2. _Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics_,
         Devaney.  in Mathematics, Addison-Wesley:  Menlo Park, CA, 1990
         This is elementary and readable, yet rigorous.
     3. _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.

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

Subject: Nonlinearity

Q20: What is nonlinearity? What are nonlinear equations?
A20: -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?

Q21: What is a fractal? What are some examples of fractals?
A21: -A set of points whose fractal dimension exceeds its topological
     dimension.
     -Examples of fractals:
       Sierpinski triangle, Koch snowflake, Peano curve, Mandlebrot set.
  
     Fractal References:
  
     1. Rudy Rucker's _Mind Tools_
     Plus many of the books mentioned later.


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

Subject: Fractal dimension

Q22: What is fractal dimension? How is it calculated?
     -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.
      

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

Subject: Strange attractors

Q23: What is a strange attractor?
A23: -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: Lyapunov exponents

Q24: What are Lyapunov exponents? How are they calculated?
A24: -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.

      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
      
References on self-affine fractals: (Anyone know what these are?)

Bedford, Math. Proc. Cambridge Phil. Soc. 106 (1989) 325
Falconer, Math. Proc. Cambridge Phil. Soc. 103 (1988) 339
McMullen, Nagoya Math. J. 96 (1984) 1
Przytycki & Urbanski, Studia Math. 93 (1989) 155

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

Subject: What are some other references?

_Chaos_, James Gleick (1987)
_Does God Play Dice?_, Ian Stewart (1990)
_The Beauty of Fractals_, Peitgen and Richter.
_The Fractal Geometry of Nature_, Benoit Mandlebrot
_Fractals Everywhere_, Michael Barnsley
"Mathematical Recreations" section in Scientific American.
Algorithm
Mandala
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.
"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.
_Fractal Creations_.  This is the book on 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.
_Fractals, Chaos, and Power Laws_, M. Schroeder.  This book contains a
    clearly written explanation of fractal geometry, with lots of
    puns and word play.

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

Subject: Acknowledgements

Thanks to:
Michael A. Rolenz, Murray Frank, Bruce Stewart, Kai-Mikael J, Leon Katz,
Miriam Nadel, Erik Boman, Abhijit V. Deshmukh, Tom Parker, Matt Perry,
Guy Metcalfe, Robert Hood, Gordon Erlebacher, Gerald Edgar, Duncan Foster,
Lamont Granquist, Tom Scavo, Carl Hommel, Jeffrey Shallit, Robert Drake,
Tom Menten, Jon Leech, Tal Kubo, Tommy Vaske, Ron Nelson, Gerald Edgar,
Erick Wong and others.

Special thanks to Matthew J. Bernhardt (mjb@acsu.buffalo.edu) for collecting
the chaos definitions for Q19 through Q24.

Copyright 1992 Ken Shirriff (shirriff@sprite.Berkeley.EDU)