Fractals, Chaos

Representing geometry in SecondLife
Some comments on the suitability of SecondLife as a way of presenting fractal geometry

Fractal forms found by using Google Earth
Readers are encouraged to contribute their own findings.

POVRay Fractal Raytracing Contest

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
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.
Box counting fractal dimension of volumetric data

Time exists so that not everything happens at once.

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

Fractals and Computer Graphics
Interface Magazine Article

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

Diffusion Limited Aggregation
DLA in 3D
Platonic solids
Tropical leaf
Random Attractors

Marketing idea: A potato chip in the shape of the Sierpinski gasket! It would infinitely crispy but have zero calories.

Duffing Attractor
Lorenz Attractor
Rossler Attractor
Henon Attractor
Henon Map
Henon Phase Diagrams
Star Julia
Tree roots
Attractor island
Peter de Jong
Fractal thorn

Clouds are not spheres, coastlines are not circles, bark is not smooth, nor does lightning travel in straight lines. Mandelbrot

Burning ship
Real valued mset
Insideout Dragon
Marek Dragon
Julia Set
Sin Julia
Newton Raphson
Mandelbrot powers
Q Series
Carotid function

"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

Logistic Equation
Dragon Curve
Hexagonal Gosper
Gingerbread man

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

Peano Curve
Square Sierpinski
Quadratic Gosper
Koch Curve
Quadratic Koch Island (b)
Quadratic Snowflake
Sierpinski Arrowhead
Von Koch Snowflake
Cross (a)
Cross (b)
Quadratic Koch Island (a)
Twon Dragon

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

Inverse San Marco Dragon
Grinch Dragon
Pascal Grey Code
Cooral Dragon
Tiled Dragon
Fuzzy Logic
Fractal fields
Square tile fractal
Pascal modulo
Diaxial Cantor Plane
Cross Carpet
Mandelbrot factorization
Rogers Rings
Measure Function
1D Cellular Automata
Reverse Julia
Inverse tree

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);}

IFS Spiral
IFS Mandelbrot-like
IFS tree
IFS Tree
IFS leaf
IFS Sand Dollar
IFS Tree
IFS Tree
IFS fern
IFS fern
IFS Dragon
IFS Twig
IFS star
IFS Christmas tree
IFS Curve
IFS Maple leaf
IFS Chaos Text
3D IFS bush
IFS Tile
Mauldin Gasket

Chaos: Bounded, deterministic dynamics that are aperiodic and display sensitive dependence to initial conditions.

L-System leaf
L-System bush
L-System bush
L-System bush
L-System Sticks
L-System bush
L-System bush
L-System algae
L-System algae
L-System weed
L-System bush
Computer sketching

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

Keplerian Fractals
Octahedral Sierpinski Carpet
3D IFS hedgehog
Sierpinski Carpet
3D IFS Cross
Quaternion Julia
Fractal Crown
Multifactal Mandelbrot
Kissing circles
Apollony fractal

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

Circle Squares
Circular Infinity
Boxed Set
Cross menger
Danca Fractal
Tippetts Mandelbrot
Menger in Second Life
Fractal linked tori
Fractal space filling
Gauss Map IFS
Jerusalem Menger Cube

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

Introduction to fractals

Generating noise with different power spectra laws
Including 1/f noise. Fractal planets and landscapes: Methods for creating fractal planets, landscapes, and clouds. Including frequency sythesis and midpoint displayment and a gallery of images created using Voxel World by Dmytry Lavrov.

IFS generator
Macintosh based IFS generator application and manual. Random IFS Gallery.

Sponge and Gasket
Description of two fractal objects and POVRay renderings by Angelo Pesce.

Colouring Attractors

New :   ECU Art Gallery, exhibition Oct 2014
Enhancing rock art recordings through hyperspectral photography
Mobius-Strip-Like Columnar Functional Connections are Revealed in Somato-Sensory Recepive Field Centroids
Repeatability of reconstructed models
Weld Range rock shelter: digital representations
Presenting :   High resolution imaging: Capture, storage and access. Oct 2014
Workshop: Automated 3D model reconstruction from photographs. Dec 2014
Novel imaging of heritage objects and sites. Dec 2014
Novel physical representations for the visualisation of science data and mathematics, April 2015
Attending :   eResearch 2014 (27-31 October)
Siggraph Asia 2014 (3-6 Dec)
International Conference on Virtual Systems and MultiMedia (9-12 Dec)
8th Annual International Conference on Computer Games Multimedia & Allied Technologies. (13-14 April)

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.