Fractals, Chaos, Self-Similarity

Written by Paul Bourke

The following is a collection of different explorations of fractals by the author over the years as well as explainations of various topics. It includes most known formulations, including but not limited to L-Systems, IFS (Iterated Function Systems), attractors and both 2D and 3D geometric fractals. Also included are a number of numerical solutions for computing dimensionality and other metrics.

Please note that I am available as a consultant on matters relating to fractals. This may include creating new forms, creating high quality graphics, animations, implementing algorithms (see box and compass dimension) and so on.

Random space filling of the plane
Space filling of a plane (and a line or 3D) with an arbitrary shape. Initial concept and inspiration by John Shier.

DLA (Diffusion Limited Aggregation)
Diffusion Limited Aggregation in 2D and 3D. Including constrained by a surface and a software implementation.

Fractal forms found by using Google Earth.

The geometric beauty of leaves.

Time exists so that not everything happens at once. Albert Einstein

Rendering Wada-type basins of attraction
Here's a party trick for over Christmas. Get 4 large shiny Christmas balls, some coloured wrapping paper, some lights from the Christmas tree, and you have you own fractal laboratory to amuse yourself with over the festive season.

Fractal Dimension and Self Similarity -- Box counting fractal dimension of volumetric data -- Box counting fractal dimension of point data
A particular box counting software package, Ruler or Compass Dimension, Lacunarity, Multifractal spectrum, Recurrence plots, Self Similarity. Examples of self similarity in fractals with examples from mathematics and photos of the physical world.

I wonder whether fractal images are not touching the very structure of our brains. Is there a clue in the infinitely regressing character of such images that illuminates our perception of art? Could it be that a fractal image is of such extraordinary richness, that it is bound to resonate with our neuronal circuits and stimulate the pleasure I infer we all feel? P. W. Atkins

Volumetric fractals -- Visualising volumetric fractals -- 3D volumetric fractal trajectories

Including the Bedouin, Real number Mset, Quinternion set, Sine Mset, Triternion fractal, Danca volume.

Including 1/f noise. Fractal planets and landscapes: Methods for creating fractal planets, landscapes, and clouds. Generating noise with different power spectra laws. Including frequency synthesis and midpoint displacement and a gallery of images created using Voxel World by Dmytry Lavrov. Perlin noise and turbulence: Creating arbitrary dimensional textures and other natural effects using techniques credited to Ken Perlin.

Polyhedral fractals
Sierpinski Gasket, Menger Sponge, Platonic solid fractals and their complements, Boxed set, Crossed Menger, Jerusalem Menger Cube, Cubic, Keplerian Fractals

IFS (Iterated Function Systems)
Macintosh based IFS generator application and manual. Random IFS Gallery. IFS tiles, IFS Bush, IFS Maple leaf, IFS Spiral, IFS Mandelbrot-like, IFS trees, IFS leaves, IFS Sand Dollar, IFS ferns, IFS chaos text, Dragon, Twigs, IFS hedgehog, IFS cross

L-Systems (Lindenmayer Systems)
Historical software description, examples from leaves, bushes, algae, trees, sticks and weeds, LSystem triangle, LSystem Peano, LSystem crystal, Square Sierpinski, Quadratic Gosper, Hilbert curve, LSystem board, Koch curve, Quadratic Koch Island, Quadratic Snowflake, Sierpinski arrowhead, Von Koch snowflake, LSystem cross, Pentaplexity, LSystem tiles, LSystem rings, Dragon curve, Hexagonal gosper, Krishna anklets, Mango leaf, Snake kolam.

I could be bounded in a nutshell and count myself a king of infinite space. Shakespear

Apollony fractal
Random walk
Zubieta Julia
Random Attractors
Peano Basin
Fractal thorn
Quaternion trigonometry
Gauss Map
Quaternion Julia
Marketing idea: A potato chip in the shape of the Sierpinski gasket! It would infinitely crispy but have zero calories.
Fractal linked tori
Fractal space filling
Random shell gaskets
Duffing Attractor
Lorenz Attractor
Rossler Attractor
Bouali attractor
Henon Attractor
The Pale Blue Dot from Space - Earth. "That's here. That's home. That's us. On it everyone you love, everyone you know, everyone you ever heard of, every human being who ever was, lived out their lives. The aggregate of our joy and suffering, thousands of confident religions, ideologies and economic doctrines, every hunter and forager, every hero and coward, every creator and destroyer of civilization, every king and peasant, every young couple in love, every mother and father, hopeful child, inventor and explorer, every teacher of morals, every corrupt politician, every superstar, every supreme leader, every saint and sinner in the history of our species lived there - on a mote of dust suspended in a sunbeam." Carl Sagan
Henon Map
Henon Phase Diagrams
Star Julia
Tree roots
Circle Squares
Peter de Jong
Chen Lee Attractor
Gingerbread man
Lévy curve
Clouds are not spheres, coastlines are not circles, bark is not smooth, nor does lightning travel in straight lines. Mandelbrot
Burning ship
Tippetts Mandelbrot
Inside out Dragon
Marek Dragon
Julia Set
Sin Julia
"And are you not", said Fook leaning anxiously forward, "a greater analyst than the Googleplex Star Thinker in the Seventh Galaxy of Light and Ingenuity which can calculate the trajectory of every single dust particle throughout a five-week Dangrabad Beta sand blizzard?" "A five-week sand blizzard?" said Deep Thought haughtily. "You ask this of me who have contemplated the very vectors of the atoms in the Big Bang itself? Molest me not with this pocket calculator stuff." Douglas Adams
Newton Raphson
Mandelbrot powers
Magnet fractals
Logistic Equation
Q Series
Carotid function
Computer sketching
I am so in favour of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Georg Cantor
1D Cellular Automata
Pascal modulo
Chaos Game
Square tile fractal
Grinch Dragon
Pascal Grey Code
Cooral Dragon
Polynomial attractors
N attractor
Juan attractor
Square Coral
Triple Dragon
Tsucs attractor
Mandelbrot trajectories
Bad hair day
Arneodo attractor
Is the human story largely a deterministic tale of civilisations marching to Toynbee's tune, three and a half beats to disintegration, or did the hinge of history turn on the length of Cleopatra's nose? Robert May
Juan Catrián
Fractal fields
Inverse tree
Lace attractor
Nova fractal
Cross Carpet
Mandelbrot factorization
main(k){float i,j,r,x,y=-16;while(puts(""),y++<15) for(x=0;x++<84;putchar(" .:-;!/>)|&IH%*#"[k&15])) for(i=k=r=0;j=r*r-i*i-2+x/25,i=2*r*i+y/10,j*j+i*i<11&&k++<111;r=j);}
Rogers Rings
Measure Function
Reverse Julia
Twon Dragon
Kissing circles
Mauldin Gasket
I believe the geometric proportion served the creator as an idea when He introduced the continuous generation of similar objects from similar objects. Johannes Kepler
Circular Infinity
Octahedral Sierpinski Carpet
Sierpinski Carpet
Multifactal Mandelbrot
Fractal Crown
Log Spiral
Fractals in SecondLife
PovRay contest
The real world was full of larger structures, smaller structures, simpler and more complex structures than the tiny portion comprising sentient creatures and their societies, and it required a profound myopia of scale and similarity to believe that everything beyond this shallow layer could be ignored. Greg Egan, Diaspora
Multi Julia
Z4 gasket
Cubic Julia
Fractional powers
Mandelbrot filling
Herman Rings
Reuleaux Triangle
Spiral Septagon
iabs fractal
Penrose Kite
Rational Rings
sinh() Julia
VR Attractor
Burke-Shaw attractor
Yu-Wang attractor
Chen-Celikovsky attractor
Four Wing attractor

Contributions by Gayla Chandler

Natural Fractals in Grand Canyon National Park
Fractals: A Symmetry Approach
Gasket Photography

Why is the winner of the Miss Universe contest always from planet Earth? Rob Hall

Other ...

Benfords Law
Colouring Attractors
Fractals and Computer Graphics (Historic interest only)
Introduction to fractals (Historic interest only)

In any field find the strangest thing and then explore it. John Archibald Wheeler

Contents : Papers, Presentations, Workshops
Geometry, Surfaces, Curves, Polyhedra
Fractals, Chaos, Self similarity
Domes, Planetariums, Fisheye, Spherical Mirror
Stereographics, 3D Projection
Panorama, 360 Video
Photographic Reconstruction
Miscellaneous: Projection, Modelling, Rendering
Data Formats: 3D, Audio, Image
Texture Library, Fun, Puzzles, Travel
All pages in one place
New/Updated : GoPro MAX 360 Action Cam movie layout/format
"Having a bad hair day" strange attractor
Visualising Mandelbrot Escape Trajectories
Experiments in Reverse Perspective
Tools for Spherical Mirror Projection
Offsite : FaceBook, Sketchfab, YouTube, Vimeo, Shapeways

The contents of this web site are © Copyright Paul Bourke or a third party contributor where indicated. You may print or save an electronic copy of parts of this web site for your own personal use. Permission must be sought for any other use. Any source code found here may be freely used provided credits are given to the author. Purchase of credit free licenses of material found on this site can be negotiated with the author. The author can also quote for unique variations and/or higher resolution versions of the images found on this site.