Fractals
This dossier focuses on Fractals – it includes top tips for using software to create fractals and more. This dossier has been put together by Karl Sarnow.
You can download the whole dossier in pdf format.
Background Pedagogical motivation Teaching units Students report Resources Activities Persons Printed material
Background
A fractal is a mathematical object, which contains, in a visual representation, the property of self similarity. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin “fractus” or "broken". Aside the fact that the object looks fractioned, the term fractal was given due to the strange fact that these objects have a dimension which is not a whole number but a fraction. Before Mandelbrot coined this term, the common name for such structures (the Koch snowflake, for example) was “monster curve”. Read more at: http://en.wikipedia.org/wiki/Fractal (English) http://de.wikipedia.org/wiki/Fraktal (Deutsch) http://fr.wikipedia.org/wiki/Fractal (Francais)
Pedagogical motivation
Many natural objects have a look which could be described mathematically in terms of fractals. The beauty of the calculated objects is so overwhelming, that it is easy to motivate students to learn about complex numbers, formal languages or vector functions, depending on the type of fractals you are going to use. You will find information about the following three fractal types later on: • Escape time fractals • Lindenmayer systems (Lsystems) • Iterated Function Systems (IFS) Every fractal type offers a pathway to a different topic of mathematics; one is even suited for lower secondary computer science education.
Teaching units
http://www.schulen.duesseldorf.de/gyscharnhorststr/fach/chaos/chaos.htm In the paper the authors consider the so called Verhulst Dynamic (Logistic Equation) as an example of some characteristic aspects of Nonlinear Dynamics. The material is suitable for pupils aged 1416 years. Parts of the downloadable document are in English language.
http://www.matheprisma.uniwuppertal.de/Module/Fraktal/index.htm A complete course on Iterated Function System  IFStype fractals in German language. This course requires some mathematical skills.
http://ejad.best.vwh.net/java/fractals/ An introduction to Lsystem fractals in English language. Only basic level of mathematics is required to follow the introduction.
http://icosaweb.acreunion.fr/GeomJava/abraCAda/Docs/lycee/abra11_2.pdf The PDF document in French language contains an article about using CABRI and FRACTINT in the classroom.
http://www.crdp.acgrenoble.fr/imel/lycee/delahaye/chaos/infogene.htm This site offers background information and students activities about fractals in French language.
http://www.acversailles.fr/etabliss/clgpompidouorgerus/Galerie_0304/Fractales/Explications.htm The French language pages of the College G. Pompidou contains mathematical and art contents.
Students report
http://www.fhniederrhein.de/~gkorsch/if1100/mvkm/stundenprotokoll.html This link guides to a students report from a lesson about recursion and the Sierpinski triangle in German language.
http://www.orillas.org/math/fractals.html Students from "Duiliu Zamfirescu" School in Focsani, Romania (age 1415 years) report about their work in English language.
http://www.orillas.org/math/19971998/fract.html Report about fractals in LOGO from students of School No.10 Focsani, Romania (age 1314 years).
Resources
http://wmi.math.uszeged.hu/~kovzol/XaoS/english.php A selection of fractals is shown, ready to order. Ordering prints from this source supports the development of XaoS, an open source program for the calculation and visualisation of fractals.
http://xaos.sourceforge.net/english.php The XaoS fractal program, available in versions for Windows, Mac OS X and Linux. Tips of how to compile a Mac OS X version are available. Excellent tutorial material comes with the program. The program is contained in many Linux distributions. The tutorials are available in English, French, Czech, German, Spanish and Hungarian language.
http://www.fhlueneburg.de/mathelehramt/fraktale/weglinde/weglinde.htm Introduction into Lsystems fractals for students of mathematics teaching in German language.
http://spanky.triumf.ca/www/fractint/fractint.html The web pages of Fractint, a free fractal generating program available for Windows, Mac and Linux.
http://www.fraktalwelt.de Information about programs to create a variety of fractal types in English and German language. Programs run under Windows only.
http://www.javaview.de/vgp/tutor/lsystem/PaLSystem.html A JAVA program (platform independent) to calculate and display 2D Lsystems online. No installation needed.
http://perso.wanadoo.fr/charles.vassallo/index.html Deep information about the mathematics and aesthetic aspects of fractals available in French and English language.
http://www.csvt.qc.ca/patriotes/pei/travaux/fractales/fractales.html A French language introduction to fractals.
http://www.ajlogo.com/ The page has a wonderful and easy to use interface to the LOGO programming language. Even a chat is available while programming. The program is a JAVA applet and therefore running directly in the Internet browser on all platforms with a JAVA plugin installed.
http://www.cs.berkeley.edu/~bh/logo.html University of California in Berkley has a LOGO version ready for free download. It is available for UNIX/LINUX, Windows and Mac.
http://xlogo.free.fr/ A free JAVA implementation of LOGO. Due to the JAVA implementation, this LOGO version runs on all platforms. The Koch coastal line is contained in the documentation as an example. The documentation is available in French, English and Spanish.
http://www.fractalsciencekit.com/ The Fractal Science Kit fractal generator provides a rich framework for exploring the world of fractals and creating your own fractal image.
Activities
http://www.fractalartcontests.com/1997/contest.htm The web page of fractal contests realized by the developers of Fractint.
Persons
http://www.math.yale.edu/mandelbrot/ The homepage of Benoit B. Mandelbrot, one of the fathers of fractal research and Xplora patron.
http://pages.cpsc.ucalgary.ca/~pwp/ The home page of PRZEMYSLAW PRUSINKIEWICZ, one of the authors of the famous book “The algorithmic beauty of plants” about Lsystems [2].
http://wwwmaths.anu.edu.au/~barnsley/ The home page of Michael Barnsley, who has drawn the IFS into the public. Michael Barnsley is the author of “Fractals everywhere” [3].
Printed material
[1] The Fractal Geometry of Nature, Benoit B. Mandelbrot, ISBN: 0716711869, 1982. [2] The Algorithmic Beauty of Plants, Prusinkiewicz & Lindenmayer, ISBN: 0387972978, 1990 (Out of print). PDF version at: http://algorithmicbotany.org/papers/abop/abop.pdf. [3] Fractals Everywhere, Michael F. Barnsley, ISBN: 0120790696, 1993.
