Fractals, Chaos, SelfSimilarity
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 LSystems,
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 Wadatype 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.
Noise
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 Mandelbrotlike,
IFS trees, IFS leaves, IFS Sand Dollar, IFS ferns, IFS chaos text,
Dragon, Twigs, IFS hedgehog, IFS cross
LSystems (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

Popcorn

Zubieta Julia

Random Attractors

Peano Basin

Fractal thorn

Wallpaper

Quaternion Julia

Hypercomplex

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

Wada

Random shell gaskets

Solenoid

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

Clifford

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

Lemon

Inside out Dragon

Marek Dragon

Buddhabrot

Mandelbrot

Julia Set

Sin Julia

Cactus

"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 fiveweek Dangrabad Beta sand blizzard?"
"A fiveweek 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

Biomorphs

Mandelbrot powers

Binocular

Magnet fractals

Logistic Equation

Oneness

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

Stretch/fold

GumowskiMira

Grinch Dragon

Ikeda

Pascal Grey Code

Cooral Dragon

Polynomial attractors





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

Curlicue

Trianguloid

Fractal fields

Inverse tree

Lace attractor

Nova fractal

Tetration

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*ri*i2+x/25,i=2*r*i+y/10,j*j+i*i<11&&k++<111;r=j);}

Rogers Rings

Measure Function

Trifaction

Woggle

Reverse Julia

Multifractal

Twon Dragon

Gauss Map

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

Star

Curve

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

Z^{4} gasket

Cubic Julia

Fractional powers

Mandelbrot filling

Rings

Herman Rings

Reuleaux Triangle

Spiral Septagon

iabs fractal

Penrose Kite

Rational Rings

sinh() Julia

Lace

BioBlobs






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

New/Updated :

Creating unwrapped icosahedral maps
Various cartographic mappings applied to equirectangular images
Experiments in Reverse Perspective
The growth of cognition: free energy minimization and the embryogenesis of cortical computation.
Realtime spectral radiance estimation of hemispherical clear skies ...
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.
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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
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