Fractals, Chaos, Self-Similarity

The following is a collection of fractal, chaos and attractors by the author. 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.

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.

The geometric beauty of leaves.

Fractal forms found by using Google Earth.

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.

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 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

3D Rendering of the Quaternion Mandelbrot Set with Memory. Fractals, Complex Geometry, Patterns, and Scaling in Nature and Society, World Scientific, published April 2023.

Calculating Julia fractal sets in any embedding dimension. Fractals, Complex Geometry, Patterns, and Scaling in Nature and Society, World Scientific, published February 2023.

Graphical Analysis of Attraction Basins of Chaotic Hidden Attractors: Rabinovich-Fabrikant System. International Journal of Bifurcation and Chaos.

Buddhabrot	Tippetts Mandelbrot	Mandelbrot	Quadratic Mandelbrot	Mandelbulb
Mandelbrot powers	Mandelbrot trajectories	Mandelbrot factorization	Multifactal Mandelbrot	Mandelbrot filling
Cactus	Marek Dragon	Burning ship	Lemon	Inside out Dragon
Clouds are not spheres, coastlines are not circles, bark is not smooth, nor does lightning travel in straight lines. Mandelbrot
Star Julia	Quaternion Julia	Zubieta Julia	Julia Set	Sin Julia
Cubic Julia	Multi Julia	Cubic Julia	sinh() Julia	Reverse Julia
Alexanders Horns	Symmetry in Chaos	Mosely Snowflake	samila	Pelham fractal
mrly fractals	Brocolli	Antoines Necklace	Simones Orbits	Midjourney fractals
It takes considerable knowledge just to realize the extent of your own ignorance. Thomas Sowell
Duffing Attractor	Lorenz Attractor	Rossler Attractor	Bouali attractor	Henon Attractor
Chen Lee Attractor	Polynomial attractors	N attractor	Juan attractor	Random Attractors
VR Attractor	Burke-Shaw attractor	Yu-Wang attractor	Chen-Celikovsky attractor	Four Wing attractor
Tsucs attractor	Sprott attractor	Arneodo attractor	Lace	Wallpaper
Peter de Jong	Clifford	Henon Map	Gingerbread man	Henon Phase Diagrams
Gumowski-Mira	Grinch Dragon	Ikeda	Cooral Dragon	Twon Dragon
Marketing idea: A potato chip in the shape of the Sierpinski gasket! It would infinitely crispy but have zero calories.
Quintonion	Quaternion trigonometry	Gauss Map	BioBlobs	Hypercomplex
Bedouin	Danca volumetric	Random shell gaskets	Solenoid	Fractal space filling
Fractal linked tori	Wada	PovRay contest	Fractals in SecondLife	Simone Conradi
"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
Apollony fractal	Random walk	Popcorn	Lévy curve	Pascal Grey Code
Newton Raphson	Bad Newton	Biomorphs	Circle Squares
Nova fractal			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
Square Coral	Triple Dragon		Lace attractor	Phoenix Julia
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
Z4 gasket		Tetration	Cross Carpet	Fractional powers
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	Trifaction	Woggle	Bad hair day
Multifractal	Mauldin Gasket	Lambda fractal	Kissing circles	Barnsleys Tree
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
Log Spiral	Fractal Crown	Spiral Julia	Polygon IFS	Percolation
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
Rings	Herman Rings	Reuleaux Triangle	Spiral Septagon	iabs fractal
Penrose Kite	Rational Rings	Tree roots	Peano Basin	Fractal thorn

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

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) Recent: Cultural Big Data: Nineteenth to Twenty-first Century Panoramic Visualization − Cortical Development in the Structural Model and Free Energy Minimization − Digitising the Murten Panorama. The story behind the highest resolution scan of an artwork − Capturing high resolution stereoscopic panorama images with a single camera − Removing tourists from photographs by averaging − Focus Peaking - An Algorithm − Vertical Panoramas − Symmetry in Chaos − Dome Projection and Calibration for Offaxis Fisheye Lenses − Tools for Spherical Mirror Projection Copyright: 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 personal use, permission must be sought for any other use.