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Fractals are one of the most interesting puzzles of mathematics. Many fractals can be made by a simple formula, yet they have such beautiful and complex designs. This page provides simple Mathematica code for many of the most interesting types of fractals. Some of this code may not be very efficient, but it should be enough to give you a basic understanding of the mathematics involved.
Kleinian Double 1/15 Cusp Group - calculated in Mathematica 4.2, rendered in POV-Ray 3.6.1, 8/10/05
Kleinian Quasifuchsian Limit Set - new version: C#, old version: Mathematica 4.2, POV-Ray 3.6.1, 4/15/06
"Mandelbrot Pearls" (Mandelbrot Set Orbits), calculated in Mathematica 4.2, rendered in POV-Ray 3.6.1, 12/24/09
Magnetic Pendulum Strange Attractor Mathematica 4.2 version: 1/24/05; C++ version: 10/27/07; 3D rendered in POV-Ray 3.1
Mandelbrot Set, using Escape Time Algorithm (ETA) - original version: Java, 5/24/01; animated version: C++, 2/16/05
Nebulabrot - Mathematica 4.2, 8/3/04; C++ version: 5/16/07
Polynomial Roots - Mathematica 4.2, 12/29/08
2D Tree Fractal - POV-Ray version: 8/20/10, Mathematica version: 10/17/04
3D Tree Fractal - POV-Ray 3.6.1, 8/20/10
Volumetric 3D Tree Fractal - C++, C# 2005 Express Edition, 8/20/10
Diffusion Limited Aggregation (DLA) Fractal - Mathematica 4.2, 8/12/04; C++ version: 2/10/05
Weierstrass Function, POV-Ray 3.6.1: 5/18/08, Mathematica 4.2: 3/2004
Romanesco Broccoli, 2/29/12
Recursive (7,3) Poincaré Hyperbolic Disk - AutoLisp, POV-Ray 3.6.1, C#, 5/17/11
Wada Basin - Mathematica 4.2, 6/19/05; C++ version: 9/11/08
“The Watering Hole” Orbit Trap - old version: Mathematica 4.2, 2/26/05; animated version: C++, 7/12/06
Gears Orbit Trap - POV-Ray 3.1, C++, 7/12/06
Mandelbrot Set Tessellation - Mathematica 4.2, 6/14/04
Golden Ratio Spiral Orbit Trap - Mathematica 4.2, 3/3/05
Newton-Raphson Fractal - Mathematica 4.2, 6/17/04
Julia Set Fractal - Mathematica 4.2: 4/15/04, POV-Ray 3.6.1: 7/4/06
Cubic Julia Set Fractal - original version: Java, 6/9/01; animated version: Mathematica 4.2, 2/18/05
Distance Estimation - Mathematica 4.2, 11/1/09
Inverse Julia Set Fractal - Mathematica 4.2, 6/26/04
Inverse Mandelbrot Set Fractal, 12/22/09
Glynn Julia Set Fractal, Mathematica 4.2, 10/13/09
Burning Ship Fractal - Mathematica 4.2, 8/5/04
Barnsley’s Fern - Mathematica 4.2, 10/16/04
Flame Fractal - Mathematica 4.2, 3/5/05
Clifford Attractor - adapted from Paul Richards, Mathematica 4.2, 12/27/04
Frequency Filtered Random Noise - Mathematica 4.2, 9/5/04
Perlin Noise - Mathematica 4.2, 8/16/04
Apollonian Gasket - Mathematica 4.2, 7/20/05
3D Apollonian Fractal - ApolFrac, 8/12/09
Hénon Map Escape Time - as seen on MathWorld, Mathematica 4.2, 6/16/04
Lorenz Attractor - POV-Ray 3.6.1, 4/10/06
Mandelbrot Set Pickover Stalks - Mathematica 4.2, 2/24/05
Quasifuchsian Limit Set - Mathematica 4.2, POV-Ray 3.6.1, 4/15/06
Crackle Fractal - Mathematica 4.2, 11/17/04
Mandelbrot Set Height Field - Mathematica 4.2, MathGL3d, 10/17/04
Fractal Crown - as described on Paul Bourke’s web site, equations by Roger Bagula - Mathematica 4.2, 8/13/04
Bifurcation Diagram (Feigenbaum Fractal) - Mathematica 4.2, 10/17/02
Circular Orbit Trap - Mathematica 4.2, 2/24/05
Want to see more? Click here to see more fractals along with Mathematica code:
Fractals LinksAnnual Fractal Art Contest Felix Hausdorff - the Hausdorff dimension. He was a Jewish scientist who sadly commited suicide to avoid being taken to concentration camp. Subdivided Columns - formed from 2700 slices of laser cut cardboard, by Michael Hansmeyer The Art of Mathematics - fractal slideshow narrated by Lasse Rempe Summary of Fractal Types - a long list by Noel Giffin |