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of Mind : In Search of a Symmetry Bond (Advances in Consciousness
Research, No 7)_, John Benjamins, ISBN: 1556191871, Subjects include:
Neural networks (Neurobiology), Mathematical models, Fractals, and
Consciousness
G.V. Middleton, (ed), _1991: Nonlinear Dynamics, Chaos and Fractals
(w/ application to geological systems)_ Geol. Assoc. Canada, Short
Course Notes Vol. 9, 235 p. This volume contains a disk with some
examples (also as pascal source code) ($25 CDN)
T.F. Nonnenmacher, G.A Losa, E.R Weibel (ed.) _Fractals in Biology and
Medicine_ ISBN 0817629890, Springer Verlag, 1994
L. Nottale, _Fractal Space-Time and Microphysics, Towards a Theory of
Scale Relativity_, World Scientific (1993).
E. Ott, _Chaos in dynamical systems_, Cambridge University Press,
1993.
E. Ott, T. Sauer, J.A. Yorke (ed.) _Coping with chaos : analysis of
chaotic data and the exploitation of chaotic systems_, New York, J.
Wiley, 1994.
D. Peak and M. Frame, _Chaos Under Control: The Art and Science of
Complexity_, W.H. Freeman and Company, New York 1994, ISBN
0-7167-2429-4 "The book is written at the perfect level to help a
beginner gain a solid understanding of both basic and subtler appects
of chaos and dynamical systems." - a review from the back cover
H. O. Peitgen and P. H. Richter, _The Beauty of Fractals_,
Springer-Verlag, New York, 1986. ISBN 0-387-15851-0. This book has
lots of nice pictures. There is also an appendix giving the
coordinates and constants for the color plates and many of the other
pictures.
H. Peitgen and D. Saupe, eds., _The Science of Fractal Images_,
Springer-Verlag, New York, 1988. ISBN 0-387-96608-0. This book
contains many color and black and white photographs, high level math,
and several pseudocoded algorithms.
H. Peitgen, H. Juergens and D. Saupe, _Fractals for the Classroom_,
Springer-Verlag, New York, 1992. These two volumes are aimed at
advanced secondary school students (but are appropriate for others
too), have lots of examples, explain the math well, and give BASIC
programs.
H. Peitgen, H. Juergens and D. Saupe, _Chaos and Fractals: New
Frontiers of Science_, Springer-Verlag, New York, 1992.
E. Peters, _Fractal Market Analysis - Applying Chaos Theory to
Investment & Economics_, John Wiley & Sons, 1994, ISBN 0-471-58524-6.
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.
C. Pickover, _Keys to Infinity_, (1995) John Wiley: NY. ISBN
0-471-11857-5.
C. Pickover, (1995) _Chaos in Wonderland: Visual Adventures in a
Fractal World._ St. Martin's Press: New York. ISBN 0-312-10743-9.
(Devoted to the Lyapunov exponent.)
C. Pickover, _Computers and the Imagination_ (Subtitled: Visual
Adventures from Beyond the Edge) (1993) St. Martin's Press: New York.
C. Pickover. _The Pattern Book: Fractals, Art, and Nature_ (1995)
World Scientific. ISBN 981-02-1426-X Some of the patterns are
ultramodern, while others are centuries old. Many of the patterns are
drawn from the universe of mathematics.
C. Pickover, _Visualizing Biological Information_ (1995) World
Scientific: Singapore, New Jersey, London, Hong Kong.
on the use of computer graphics, fractals, and musical techniques to
find patterns in DNA and amino acid sequences.
C. Pickover, _Fractal Horizons: The Future Use of Fractals._ (1996)
St. Martin's Press, New York.
Speculates on advances in the 21st Century. Six broad sections:
Fractals in Education, Fractals in Art, Fractal Models and Metaphors,
Fractals in Music and Sound, Fractals in Medicine, and Fractals and
Mathematics. Topics include: challenges of using fractals in the
classroom, new ways of generating art and music, the use of fractals
in clothing fashions of the future, fractal holograms, fractals in
medicine, fractals in boardrooms of the future, fractals in chess.
J. Pritchard, _The Chaos Cookbook: A Practical Programming Guide_,
Butterworth-Heinemann, Oxford, 1992. ISBN 0-7506-0304-6. It contains
type in and go listings in BASIC and Pascal. It also eases you into
some of the mathematics of fractals and chaos in the context of
graphical experimentation. So it's more than just a
type-and-see-pictures book, but rather a lab tutorial, especially good
for those with a weak or rusty (or even nonexistent) calculus
background.
P. Prusinkiewicz and A. Lindenmayer, _The Algorithmic Beauty of
Plants_, Springer-Verlag, NY, 1990. ISBN 0-387-97297-8. A very good
book on L-systems, which can be used to model plants in a very
realistic fashion. The book contains many pictures.
Edward R. Scheinerman, _Invitation to Dynamical Systems_,
Prentice-Hall, 1996, ISBN 0-13-185000-8, xvii + 373 pages
M. Schroeder, _Fractals, Chaos, and Power Laws: Minutes from an
Infinite Paradise_, W. H. Freeman, New York, 1991. This book contains
a clearly written explanation of fractal geometry with lots of puns
and word play.
J. Sprott, _Strange Attractors: Creating Patterns in Chaos_, M&T Books
(subsidary of Henry Holt and Co.), New York. ISBN 1-55851-298-5. This
book describes a new method for generating beautiful fractal patterns
by iterating simple maps and ordinary differential equations. It
contains over 350 examples of such patterns, each producing a
corresponding piece of fractal music. It also describes methods for
visualizing objects in three and higher dimensions and explains how to
produce 3-D stereoscopic images using the included red/blue glasses.
The accompanying 3.5" IBM-PC disk contain source code in BASIC, C,
C++, Visual BASIC for Windows, and QuickBASIC for Macintosh as well as
a ready-to-run IBM-PC executable version of the program. Available for
$39.95 + $3.00 shipping from M&T Books (1-800-628-9658).
D. Stein (ed), _Proceedings of the Santa Fe Institute's Complex
Systems Summer School_, Addison-Wesley, Redwood City, CA, 1988. See
especially the first article by David Campbell: "Introduction to
nonlinear phenomena".
R. Stevens, _Fractal Programming in C_, M&T Publishing, 1989 ISBN
1-55851-038-9. This is a good book for a beginner who wants to write a
fractal program. Half the book is on fractal curves like the Hilbert
curve and the von Koch snow flake. The other half covers the
Mandelbrot, Julia, Newton, and IFS fractals.
I. Stewart, _Does God Play Dice?: the Mathematics of Chaos_, B.
Blackwell, New York, 1989.
Y. Takahashi, _Algorithms, Fractals, and Dynamics_, Plenum Pub Corp,
(May) 1996, ISBN: 0306451271 Subjects: Differentiable dynamical syste,
Congresses, Fractals, Algorithms, Differentiable Dynamical Systems,
Algorithms (Computer Programming)
T. Wegner and B. Tyler, _Fractal Creations_, 2nd ed. The Waite Group,
1993. ISBN 1-878739-34-4 This is the book describing the Fractint
program.
_Q29b_: What are some relevant journals?
_A29b_: Some relevant journals are:
"Chaos and Graphics" section in the quarterly journal _Computers and
Graphics_. This contains recent work in fractals from the graphics
perspective, and usually contains several exciting new ideas.
"Mathematical Recreations" section by I. Stewart in _Scientific
American_.
"Fractal Trans-Light News" published by Roger Bagula
(). Roger Bagula 11759 Waterhill Road, Lakeside,
CA 92040 USA. Fractal Trans-Light News is a newsletter of mathematics,
computer programs, art and poetry. To subscribe, send USD $20 (USD $50
for overseas delivery) to the address above.
_Fractal Report_. Reeves Telecommunication Labs.
West Towan House, Porthtowan, TRURO, Cornwall TR4 8AX, U.K.
WWW: http://ourworld.compuserve.com/homepages/JohndeR/fractalr.htm
Email: John@longevb.demon.co.uk (John de Rivaz)
_FRAC'Cetera_. This is a gazetteer of the world of fractals and
related areas, supplied on IBM PC format HD disk. FRACT'Cetera is the
home of FRUG - the Fractint User Group. For more information, contact:
Jon Horner, Editor,
FRAC'Cetera Le Mont Ardaine, Rue des Ardains, St. Peters Guernsey GY7
9EU Channel Islands, United Kingdom. Email: 100112.1700@compuserve.com
_Fractals, An interdisciplinary Journal On The Complex Geometry of
Nature
_This is a new journal published by World Scientific. B.B Mandelbrot
is the Honorary Editor and T. Vicsek, M.F. Shlesinger, M.M Matsushita
are the Managing Editors). The aim of this first international journal
on fractals is to bring together the most recent developments in the
research of fractals so that a fruitful interaction of the various
approaches and scientific views on the complex spatial and temporal
behavior could take place.
_________________________________________________________________
_Q28c_: What are some other Internet references?
_A28c_: Some other Internet references:
Web references to nonlinear dynamics
Dynamical Systems (G. Zito)
http://alephwww.cern.ch/~zito/chep94sl/sd.html
Scanning huge number of events (G. Zito)
http://alephwww.cern.ch/~zito/chep94sl/chep94sl.html
The Who Is Who Handbook of Nonlinear Dynamics
http://www.nonlin.tu-muenchen.de/chaos/Dokumente/WiW/wiw.html
Multifractals
_Q30_: What are multifractals?
_A30_: It is not easy to give a succinct definition of multifractals.
Following Feder (1988) one may distinguish a measure (of probability,
or some physical quantity) from its geometric support - which might or
might not have fractal geometry. Then if the measure has different
fractal dimension on different parts of the support, the measure is a
multifractal.
Hastings and Sugihara (1993) distinguish multifractals from
multiscaling fractals - which have different fractal dimensions at
different scales (e.g. show a break in slope in a dividers plot, or
some other power law). I believe different authors use different names
for this phenomenon, which is often confused with true multifractal
behaviour.
Aliasing
_Q31a_: What is aliasing?
_A31a_: In computer graphics circles, "aliasing" refers to the
phenomenon of a high frequency in a continuous signal masquerading as
a lower frequency in the sampled output of the continuous signal. This
is a consequence of the discrete sampling used by the computer.
Put another way, it is the appearance of "chuckiness" in an still
image. Because of the finite resolution of a computer screen, a single
pixel has an associate width, whereas in mathematics each point is
infintesimely small, with _no width_. So a single pixel on the screen
actually visually represents an infinite number of mathematical
points, each of which may have a different correct visual
representation.
_Q31b_: What does aliasing have to do with fractals?
_A31b_: Fractals, are very strange objects indeed. Because they have
an infinite amount of arbitrarily small detail embedded inside them,
they have an infinite number of frequencies in the images. When we use
a program to compute an image of a fractal, each pixel in the image is
actually a sample of the fractal. Because the fractal itself has
arbitrarily high frequencies inside it, we can never sample high
enough to reveal the "true" nature of the fractal. _Every_ fractal
ever computed has aliasing in it. (A special kind of aliasing is
called "Moire' patterns" and are often visible in fractals as well.)
_Q31c_: How Do I "Anti-Alias" Fractals?
_A31c_: We can't eliminate aliasing entirely from a fractal but we can
use some tricks to reduce the aliasing present in the fractal. This is
what is called "anti-aliasing." The technique is really quite simple.
We decide what size we want our final image to be, and we take our
samples at a higher resolution than our final size. So if we want a
100x100 image, we use at least 3 times the number of pixels in our
"supersampled" image - 300x300, or 400x400 for even better results.
But wait, we want a 100x100 image, right? Right. So far, we haven't
done anything special. The anti-aliasing part comes in when we take
our supersampled image and use a filter to combine several adjacent
pixels in our supersampled image into a single pixel in our final
image. The choice of the filter is very important if you want the best
results! Most image manipulation and paint programs have a resize with
anti-aliasing option. You can try this and see if you like the
results. Unfortunately, most programs don't tell you exactly what
filter they are applying when they "anti-alias," so you have to
subjectively compare different tools to see which one gives you the
best results.
The most obvious filter is a simple averaging of neighbouring pixels
in the supersampled image. Being the most obvious choice, it is
generally the one most widely implemented in programs. Unfortunately
it gives poor results. However, many fractal programs are now
beginning to incorporate anti-aliasing directly in the fractal
generation process along with a high quality filter. Unless you are a
programmer, your best bet is to take your supersampled image and try
different programs and filters to see which one gives you the best
results.
An example of such filtering in a fractal program can be found on
Dennis C. De Mars' web page on anti-aliasing in his FracPPC program:
http://members.aol.com/dennisdema/anti-alias/anti-alias.html
References
The original submission from Rich Thomson is available from
http://www.mta.ca/~mctaylor/fractals/aliasing.html
To read more about Digital Signal Processing, a good but technical
book is "Digital Signal Processing", by Alan V. Oppenheim and Ronald
W. Schafer, ISBN 0-13-214635-5, Prentice-Hall, 1975.
For more on anti-aliasing filters and their application to computer
graphics, you can read "Reconstruction Filters in Computer Graphics",
Don P. Mitchell, Arun N. Netravali, Computer Graphics, Volume 22,
Number 4, August 1988. (SIGGRAPH 1988 Proceedings).
If you're a programmer type and want to experiment with lots of
different filters on images, or if you're looking for an efficient
sample implementation of digital filtering, check out Paul Heckbert's
zoom program at ftp://ftp.cs.utah.edu/pub/painter/zoom.tar.gz
Science Fair Projects
_Q32_: Ideas for science fair projects?
_A32_: You should check with your science teacher about any special
rules and restrictions. Fractals are really an area of mathematics and
mathematics may be a difficult topic for science fairs with an
experimental bias.
1. Modelling real-world phenomena with fractals, e.g. Lorenz's
weathers models or fractal plants and landscapes
2. Calculate the fractal (box-counting) dimension of a leaf, stone,
river bed
3. _How long is a coastline?_, see The Fractal Geometry of Nature
4. Check books and web sites aimed at high school students.
Subject: Notices
_Q33_: Are there any special notices?
_A33_:
From: Lee Skinner
Date: Sun, 26 Oct 1997 12:37:33 -0500
Subject: Explora Science Exhibit
Explora Science Exhibit
The newly combined Explora Science Center and Children's Museum of
Albuquerque had its Grand Opening on Saturday October 25 1997. One of
the best exhibits is one illustrating fractals and fractal art.
Posters made by Doug Czor illustrate how fractals are computed.
Fractal-art images were exhibited by Lee Skinner, Jon Noring, Rollo
Silver and Bob Hill. The exhibit will probably be on display for about
6 months. Channel 13 News had a brief story about the opening and
broadcasted some of the fractal-art images. The museum's gift shop is
selling Rollo's Fractal Universe calendars and 4 different mouse-pad
designs of fractals by Lee Skinner. Two of the art pieces are
18432x13824/65536 Cibachrome prints using images recalculated by Jon
Noring.
Lee Skinner
_________________________________________________________________
From: Javier Barrallo
Date: Sun, 14 Sep 1997 18:06:14 +0200
Subject: Mathematics & Design - 98
INVITATION AND CALL FOR PAPERS
Second International Conference on Mathematics & Design 98
Dear friend,
This is to invite you to participate in the Second International
Conference on Mathematics & Design 98 to be held at San Sebastian,
Spain, 1-4 June 1998.
The main objective of these Conferences is to bring together
mathematicians, engineers, architects, designers and scientists
interested on the interaction between Mathematics and Design, where
the world design is understood in its more broad sense, including all
types of design.
Further information and a regularly updated program is available
under:
http://www.sc.ehu.es/md98
We will be pleased if you kindly forward this message to colleagues of
yours who might be interested in this announcement.
Hoping to be able to have your valuable collaboration and assistance
to the Conference,
The Organising Committee
E-mail: mapbacaj@sa.ehu.es
_________________________________________________________________
From: John de Rivaz
Mr Roger Bagula, publisher of The Fractal Translight Newsletter, is seeking
new articles. Write to him for a sample copy - he is not on the Internet -
and he appreciates something for materials and postage.
Mr Roger Bagula,
11759 Waterhill Road
Lakeside
CA 90240-2905
USA
_________________________________________________________________
NOTICE from J. C. (Clint) Sprott :
The program, Chaos Data Analyzer, which I authored is a research and
teaching tool containing 14 tests for detecting hidden determinism in
a seemingly random time series of up to 16,382 points provided by the
user in an ASCII data file. Sample data files are included for model
chaotic systems. When chaos is found, calculations such as the
probability distribution, power spectrum, Lyapunov exponent, and
various measures of the fractal dimension enable you to determine
properties of the system Underlying the behavior. The program can be
used to make nonlinear predictions based on a novel technique
involving singular value decomposition. The program is menu-driven,
very easy to use, and even contains an automatic mode in which all the
tests are performed in succession and the results are provided on a
one-page summary.
Chaos Data Analyzer requires an IBM PC or compatible with at least
512K of memory. A math coprocessor is recommended (but not required)
to speed some of the calculations. The program is available on 5.25 or
3.5" disk and includes a 62-page User's Manual. Chaos Data Analyzer is
peer-reviewed software published by Physics Academic Software, a
cooperative Project of the American Institute of Physics, the American
Physical Society, And the American Association of Physics Teachers.
Chaos Data Analyzer and other related programs are available from The
Academic Software Library, North Carolina State University, Box 8202,
Raleigh, NC 27695-8202, Tel: (800) 955-TASL or (919) 515-7447 or Fax:
(919) 515-2682. The price is $99.95. Add $3.50 for shipping in U.S. or
$12.50 for foreign airmail. All TASL programs come with a 30-day,
money-back guarantee.
_________________________________________________________________
From Clifford Pickover
You are cordially invited to submit interesting, well-written articles
for the "Chaos and Graphics Section" of the international journal
Computers and Graphics. I edit this on-going section which appears in
each issue of the journal. Topics include the mathematical,
scientific, and artistic application of fractals, chaos, and related.
Your papers can be quite short if desired, for example, often a page
or two is sufficient to convey an idea and a pretty graphic. Longer,
technical papers are also welcome. The journal is peer-reviewed. I
publish color, where appropriate. Write to me for guidelines. Novelty
of images is often helpful.
Goals
The goal of my section is to provide visual demonstrations of
complicated and beautiful structures which can arise in systems based
on simple rules. The section presents papers on the seemingly
paradoxical combinations of randomness and structure in systems of
mathematical, physical, biological, electrical, chemical, and artistic
interest. Topics include: iteration, cellular automata, bifurcation
maps, fractals, dynamical systems, patterns of nature created from
simple rules, and aesthetic graphics drawn from the universe of
mathematics and art.
Subject: Acknowledgements
_Q34_: Who has contributed to the sci.fractals FAQ?
_A34_: Former editors, participants in the Usenet group sci.fractals
and the listserv forum frac-l have provided most of the content of
sci.fractals FAQ. For their help with this FAQ, "thank you" to:
Alex Antunes, Donald Archer, Simon Arthur, Roger Bagula, John Beale,
Matthew J. Bernhardt, Steve Bondeson, Erik Boman, Jacques Carette,
John Corbit, Douglas Cootey, Charles F. Crocker, Michael Curl, Predrag
Cvitanovic, Paul Derbyshire, John de Rivaz, Abhijit Deshmukh, Tony
Dixon, Jürgen Dollinger, Robert Drake, Detlev Droege, Gerald Edgar,
Glenn Elert, Gordon Erlebacher, Yuval Fisher, Duncan Foster, David
Fowler, Murray Frank, Jean-loup Gailly, Noel Giffin, Frode Gill, Terry
W. Gintz, Earl Glynn, Lamont Granquist, John Holder, Jon Horner, Luis
Hernandez-Urëa, Jay Hill, Arto Hoikkala, Carl Hommel, Robert Hood,
Larry Husch, Oleg Ivanov, Henrik Wann Jensen, Simon Juden, J.
Kai-Mikael, Leon Katz, Matt Kennel, Robert Klep, Dave Kliman, Pavel
Kotulsky, Tal Kubo, Per Olav Lande, Paul N. Lee, Jon Leech, Otmar
Lendl, Ronald Lewis, Jean-Pierre Louvet, Garr Lystad, Jose Oscar
Marques, Douglas Martin, 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, Olaf
G. Podlaha, Francesco Potortì, Kevin Ring, Michael Rolenz, Tom Scavo,
Jeffrey Shallit, Ken Shirriff, Rollo Silver, Lee H Skinner, David
Sharp, J. C. Sprott, Gerolf Starke, Bruce Stewart, Dwight Stolte,
Michael C. Taylor, Rich Thomson, Tommy Vaske, Tim Wegner, Andrea
Whitlock, David Winsemius, Erick Wong, Wayne Young, Giuseppe Zito, and
others.
A special thanks to Jean-Pierre Louvet, who has taken on the task of
maintaining the sections for fractal software and where fractal
pictures are archived.
If I have missed you, I am very sorry, let me know
(fractal-faq@mta.ca) and I will add you to the list. Without the help
of these contributors, the sci.fractals FAQ would be not be possible.
Subject: Copyright
_Q35_: Copyright?
_A35_: This document, "sci.fractals FAQ", is _Copyright © 1997-1998 by
Michael C. Taylor and Jean-Pierre Louvet._ All Rights Reserved. This
document is published in New Brunswick, Canada.
Previous versions:
Copyright 1995-1997 Michael Taylor
Copyright 1995 Ermel Stepp (edition v2n1)
Copyright 1993-1994 Ken Shirriff
The Fractal FAQ was created by Ken Shirriff and edited by him through
September 26, 1994. The second editor of the Fractal FAQ is Ermel
Stepp (Feb 13, 1995). Since December 2, 1995 the acting editor has
been Michael C. Taylor.
Permission is granted for _non-profit_ reproduction and distribution
of this issue of the sci.fractals FAQ as a complete document. You may
product complete copies, including this notice, of the sci.fractals
FAQ for classroom use. This _does not_ mean automatic permission for
usage in CD-ROM collections or commercial educational products. If you
would like to include sci.fractals FAQ, in whole or in part, in a
commercial product contact Michael C. Taylor.
Warranty
This document is provided as is without any express or implied
warranty.
Contacting the editors
If you would like to contact the editors, you may do so in writing at
the following addresses:
Attn: Michael Taylor
Computing Services
Mount Allison University
49A York Street
Sackville, New Brunswick E4L 1C7
CANADA
email: fractal-faq@mta.ca
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