Help commercialise these fractals!     Please email us

CLICK ON PICTURE for a high-resolution version

Fractals_2

lg_AOT18D.jpg I Saw The Stained Glass	lg_AOT18F.jpg The Eyes of the Watcher	lg_FL14001G.jpg Plastic Reality
lg_FL22001G.jpg A Carpet of Inner-Sense/Innocence	lg_FL99121C.jpg A Crisis I Knew Had To Pass	lg_FL99121E.jpg The Scale and the Anchor
lg_FLAT001D.jpg Fractured Sight	lg_morgan2.jpg The Spiral Web	lg_R981202Y.jpg Field of Caution
lg_R981215I.jpg The Interstellar Edge	lg_R981216Q.jpg This the Inner Pathway	lg_R981219H.jpg Hidden in the Deep
lg_R981220D.jpg Feminine Sky	lg_R981220S.jpg In-sight Horizon	lg_R990403H.jpg Cosmic Serf
lg_R990403T.jpg Do-Rave You Diamond Ace	lg_R990404F.jpg Suave and Unknown	lg_R990404L.jpg The Coffee-Anon
lg_R990920R.jpg Sculpture Within Scription	lg_R990920S.jpg A Soft Innocence	lg_R990920V.jpg Caught on a Wire
lg_R990920Y.jpg When Jerry Became The Grateful Dead	lg_R990921B.jpg A Magic Sight	lg_R990921E.jpg The Mark of the Canes
lg_R990921I.jpg Crystaline	lg_R990921o.jpg Something Sexual	lg_R990922A.jpg The Electric Borg
lg_R990922K.jpg The Cool Siren	lg_R990922Y.jpg A Spike of Life's Abandon	lg_R990922Z.jpg Ceremony
lg_R990923Q.jpg An Embryo of the Skies	lg_R990923R.jpg Drops of Glass	lg_R990923V.jpg The Glass Bead Game
lg_R990924F.jpg Without a License to Kill	lg_R990925Y.jpg Within the Spiral Arm	lg_R990925Z.jpg Archetype
lg_R990926N.jpg A Foreign Plane	lg_R990926Q.jpg The Mechanics of Sight	lg_R990926T.jpg Life Form Beneath the Microscope
lg_R990927B.jpg An Angel in My Wings	lg_R990927C.jpg Energy As Crystal	lg_R991202S.jpg She Sells Sanctuary

lg_STAQ003E.jpg Meditation Field	lg_STAQ003T.jpg Freedom Flow	lg_STAR001C.jpg A Centre Point
lg_STAR001F.jpg A Gentleness Within	lg_STAR001U.jpg Sun Ra	lg_STAR007R.jpg A Jewell Within an Arabian Fold
lg_STAR007T.jpg Imagination Preacher	lg_STAR008E.jpg The Temple of Life	lg_T000101R.jpg Health
lg_T000102U.jpg Hermann's Hermit	lg_T000102w.jpg The New Zeal	lg_T000103H.jpg Kaleidoscopic Twitch
lg_T000103P.jpg A Chicken For You My Dear	lg_T001101R.jpg Michael of the Viking Moor	lg_T001102O.jpg Keep All Energies You Seek
lg_T001102P.jpg Antonin Ali is the Keeper of the Gate	lg_T001102Y.jpg The Emerging Soul	lg_T001103A.jpg Converging Frills Will Keep You Satisfied
lg_T001103F.jpg Monty's Kiss	lg_T001103U.jpg Spirit of Batman's Shallow Depth	lg_T001104N.jpg The Residents Meet the Child Inside
lg_T010101I.jpg Silver Flare	lg_T010101M.jpg Point of Encryption	lg_T981201A.jpg Star Child
lg_TE0101A.jpg Embodiment of Aboriginal Youth	lg_te00011k.jpg Encoiled This Love Divine	lg_te00012e.jpg The Pea Shooter
lg_te00012h.jpg A Dextra Sinister	lg_te00012l.jpg Soft Shell	lg_TE00013A.jpg Haley's Prophet
lg_TE00013J.jpg Happy in Your Dress	lg_TE00013M.jpg I Remember the Trip You Gave	lg_TE00013Q.jpg False Breather
lg_TE00013R.jpg Beyond the Gray	lg_TE43001V.jpg The Formation of Life	lg_TE99092D.jpg This Etch of Acid
lg_TE99092J.jpg Koru	lg_TE99092O.jpg Alignment	lg_TE99092V.jpg Tourniquet
lg_TE99095Q.jpg Distant Forms	lg_TE99096M.jpg This Whispered Void	lg_WILDF01G.jpg Aztec Age

(Stunning! Commercial!) High-resolution versions available.

Please email us for any enquiries.

It is not permitted to reproduce any of these pictures without permission from the copyright-holder.
Reproductions for personal and non-commercial and non-profit use are permissible under the condition that you send a printed copy of your publication to:
Norbert HALEY
Poste Restante
P.O.Wellesley St.
Auckland 1001
New Zealand
(please include a MastHead of your local newspaper, as we collect them)

Keywords: periodic loop iteration. Complex numbers imaginary component random formula parameters . fraktal,fraktale,fraktals,fractal systems,fraktal prague,fraktalkine,fraktali,fraktal logistik,fractal video,fraktal radio,fractaline Complex plane universe self-similar and self similarity algebra julia ikenaga newton tetration tilemandel koch sierpinski hilbert space. Dragon new spiritual art mind expanding subliminal experience flower power abstract designs psychodelic pictures and meditative images beautiful graphics dramatic wild fantastic contemporary visual dream explosions neue spirtuelle computer kunst bewußtseinserweiternde unterschwellige eindrücke abstrakte designs psychodelische und meditative bilder visuelle fantastische phantanstische explosionen schöne grafik fantasie phantasie traum Fractal formulae fraktal formel mathematics imaginary numbers. Chaos theory science images geometry of nature mandelbrot amygdala frac'cetera mazes of the mind computer generated art patterns graphdev computer graphics. Imaginary value iterations bailout schönheit drucke fractal halicugenic trippy colourful computer generated art chaos apfelmännchen mandelbrot Julia geometry nature science mazes Sylvie Gallet mandel checkerboard pester markmann trapdoor ciphers Philippe Hurbain Penrose's rhombuses, generated by decomposition rules generate the fat rhombus, the thin one Individualization of rhombuses allows easy coloring. poster calendar cards Real Perturbation of Z(0) Imaginary Perturbation Real Part of Parameter Imaginary Part of Parameter Polynomial Degree Real part of Exponent Root Degree time Step Coefficient First Function term Angle (radians) Step size Stop val Points per orbit non-zero value stripes Constant imaginary Filter. Cycles Seed Population Apply mapping once color shading random Lambda Alpha Beta Gamma Omega phi kappa Degree symmetry. Function Shift Value Random seed Value Colouring method Max Hits per Pixel trig fn parameter range Evolution Mode Options randomize grid Grouting. Mutation reduction factor generations invert radius outside colour double mutation levels Variable tweak central. Fractal witchcraft algorithm hallucigenic Added Humberto Baptista's Epsilon Cross variation positive proximity value internal orbitdelay timer PI box popcorn and popcornjul Humberto Baptista Pickover's Latoocarfian Unity fractal type Chuck Ebbert's biomorph showorbit Fractint mandel julia Rees Acheson (author of MANDELB), Damien Jones, Agner Fog, Terje Mathisen, Thomas Jentzsch, and Daniele Paccaloni. Bert Tyler Stone Soup story Logarithmic Palettes and Color Ranges Biomorph Colour Decomposition Finite Attractors Continuous Potential Distance Estimator parameters corner coordinates Leibniz greek geometers Brahe Copernicus Kepler Newton ellipse parabola, hyperbola planets comets and projectiles. Calculus differentiating derivative functions geometric tangent Weierstrass Cantor Peano Poincare Benoit Mandelbrot visualisation caltech escher mind pattern clifford pickover frac'cetera complex numbers and imaginary numbers ident iteration mathematics maths stunning mind-bogling and bizarre sound maze amazing drum-n-bass rhythm rythmus musik tanz selbstaehnlich selfsimilar In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole".[1] The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured". A fractal as a geometric object generally has the following features: It has a fine structure at arbitrarily small scales. It is too irregular to be easily described in traditional Euclidean geometric language. It is self-similar (at least approximately or stochastically). It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve). It has a simple and recursive definition.[2] Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics. The mathematics behind fractals began to take shape in the 17th century when philosopher Leibniz considered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense). It took until 1872 before a function appeared whose graph would today be considered fractal, when Karl Weierstrass gave an example of a function with the non-intuitive property of being everywhere continuous but nowhere differentiable. In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition, gave a more geometric definition of a similar function, which is now called the Koch snowflake. In 1915, Waclaw Sierpinski constructed his triangle and, one year later, his carpet. Originally these geometric fractals were described as curves rather than the 2D shapes that they are known as in their modern constructions. The idea of self-similar curves was taken further by Paul Pierre Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole, described a new fractal curve, the Lévy C curve. Georg Cantor also gave examples of subsets of the real line with unusual properties—these Cantor sets are also now recognized as fractals. Iterated functions in the complex plane were investigated in the late 19th and early 20th centuries by Henri Poincaré, Felix Klein, Pierre Fatou and Gaston Julia. However, without the aid of modern computer graphics, they lacked the means to visualize the beauty of many of the objects that they had discovered. In the 1960s, Benoît Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry Richardson. Finally, in 1975 Mandelbrot coined the word "fractal" to denote an object whose Hausdorff-Besicovitch dimension is greater than its topological dimension. He illustrated this mathematical definition with striking computer-constructed visualizations. These images captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal". A relatively simple class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. Additional examples of fractals include the Lyapunov fractal and the limit sets of Kleinian groups. Fractals can be deterministic (all the above) or stochastic (that is, non-deterministic). For example, the trajectories of the Brownian motion in the plane have a Hausdorff dimension of 2. Chaotic dynamical systems are sometimes associated with fractals. Objects in the phase space of a dynamical system can be fractals (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set. Even simple smooth curves can exhibit the fractal property of self-similarity. For example the power-law curve (also known as a Pareto distribution) produces similar shapes at various magnifications. Three common techniques for generating fractals are: Escape-time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal. Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals. Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters. Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals: Exact self-similarity — This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity. Quasi-self-similarity — This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar. Statistical self-similarity — This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar. Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples — a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance" — the same electromagnetic properties no matter what the frequency — from Maxwell's equations (see fractal antenna). Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock's paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work.[4] Fractals are also prevalent in African art and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.