Fractal Art

Fractal art is created by calculating fractal objects and representing the calculation results as still images, animations, music, or other media. Fractal art is usually created indirectly with the assistance of a computer, iterating through three phases: setting parameters of appropriate fractal software, executing the possibly lengthy calculation and evaluating the product.

Fractal objects fall into four main categories depending on how an artist can manipulate their construction and rendering to exercise artistic control over the resulting fractal art:
Escape time fractals that are manipulated with the choice of the formula to be iterated and its parameters, the choice of what points are iterated (usually a tiny region of the complex plane containing interesting shapes) and how they are mapped to an image, the choice of how to compute a colour from the set of sequences of iterations. All these components have an explicitly mathematical and nonvisual nature and they can often be very complex.

Examples of this type of fractal are the Mandelbrot and Julia sets, the Lyapunov fractal, the Newton fractal, its relative the Nova fractal, and the Burning Ship fractal; some fractal image creation programs for this type of fractal art are Ultra Fractal, Gnofract4D, ChaosPro and Fractint.
Lindenmayer systems and other constructions based on replacement rules. Examples include the Peano curve and the Hilbert curve, the Sierpinski gasket and the Menger sponge, and the Koch snowflake. Stochastic systems where the replaced shapes and/or the choice of rules are random are very popular, especially to recreate trees and other natural objects.

Design relies on simple geometry (angles and lengths) and being able to predict the shapes resulting from a rule system, and the possibility of fast or realtime previews of the result greatly facilitates small adjustements of sizes, angles and probabilities.
Iterated function systems and variants thereof (fractal flames in particular); shapes and colours are determined by easily understood transformations of shrunk copies of the whole pattern, and since the transformation matrices and deformations have no particular significance they are usually input in fractal software visually and often with a realtime preview; another trend is manual editing starting from a random fractal (the arbitrary parameters are many and mostly independent). Apophysis is a popular and very sophisticated example of this category.
Stochastic synthesis of fractal noise (typically fractal landscapes) controlled through few simple high level parameters and by trying different Pseudorandom number generator seeds.

Fractals of all four kinds have been used as the basis for digital art and animation. Starting with 2-dimensional details of fractals such as the Mandelbrot Set, fractals have found artistic application in fields as varied as texture generation, plant growth simulation and landscape generation.

Fractals are sometimes combined with human-assisted evolutionary algorithms, either by iteratively choosing good-looking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing with cumbersome or unpredictable parameters, or collectively like in the Electric Sheep project, where people use fractal flames rendered with distributed computing as their screensaver and “rate” the flame they are viewing, influencing the server which reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.

Fractals in depth:

A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,” a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning “broken” or “fractured.”

A fractal often 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.

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.

Their history:

The mathematics behind fractals began to take shape in the 17th century when mathematician and 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”.

Four common techniques of generating Fractals are:

Escape-time fractals — (also known as “orbits” 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, the Nova fractal and the Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
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.
Strange Attractors — Generated by iteration of a map or the solution of a system of initial-value differential equations that exhibit chaos

The Sets in depth:

Mandelbrot Set:
In mathematics, the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial xn+1 = xn2 + c remains bounded. That is, a complex number, c, is in the Mandelbrot set if, when starting with x0=0 and applying the iteration repeatedly, the absolute value of xn never exceeds a certain number (that number depends on c) however large n gets.

Eg. c = 1 gives the sequence 0, 1, 2, 5, 26… which leads to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set.

On the other hand, c = i gives the sequence 0, i, (−1 + i), −i, (−1 + i), −i…, which is bounded, and so it belongs to the Mandelbrot set.

When computed and graphed on the complex plane, the Mandelbrot Set is seen to have an elaborate boundary, which does not simplify at any given magnification. This qualifies the boundary as a fractal.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition. Benoît Mandelbrot and others worked hard to communicate this area of mathematics to the public.

It’s History:

The Mandelbrot set has its place in complex dynamics, a field first investigated by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The first pictures of it were drawn in 1978 by Robert Brooks and Peter Matelski as part of a study of Kleinian Groups.

Mandelbrot studied the parameter space of quadratic polynomials in an article that appeared in 1980. The mathematical study of the Mandelbrot set really began with work by the mathematicians Adrien Douady and John H. Hubbard, who established many of its fundamental properties and named the set in honour of Mandelbrot.

The mathematicians Heinz-Otto Peitgen and Peter Richter became well-known for promoting the set with stunning photographs, books , and an internationally touring exhibit of the German Goethe-Institut.

The cover article of the August 1985 Scientific American introduced the algorithm for computing the Mandelbrot set to a wide audience. The cover featured an image created by Peitgen, et. al.

The work of Douady and Hubbard coincided with a huge increase in interest in complex dynamics and abstract mathematics, and the study of the Mandelbrot set has been a centerpiece of this field ever since. An exhaustive list of all the mathematicians who have contributed to the understanding of this set since then is beyond the scope of this article, but such a list would notably include Mikhail Lyubich, Curt McMullen, John Milnor, Mitsuhiro Shishikura, and Jean-Christophe Yoccoz.

Julia Set:

In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations.

The Fatou set of is the complement of the Julia set: that is, the set of points which exhibit ’stable’ behavior.

Thus on , the behavior of is ‘regular’, while on , it is ‘chaotic’.

These sets are named in honor of the French mathematicians Gaston Julia and Pierre Fatou who initiated the theory of complex dynamics in the early 20th century.

In mathematics, Lyapunov fractals (also known as Markus-Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B.

A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent λ) in the a-b plane for a given periodic sequence of as and bs. In the images, yellow corresponds to λ < 0 (stability), and blue corresponds to λ > 0 (chaos).

Properties

Lyapunov fractals are generally drawn for values of A and B in the interval [0,4]. For larger values, the interval [0,1] is no longer stable and the sequence is likely to be attracted by infinity, although convergent cycles of finite values continue to exist for some parameters. For all iteration sequences, the diagonal a = b is always the same as for the standard one parameter logistic function.

The sequence is usually started at the value 0.5, which is a critical point of the iterative function. The other (even complex valued) critical points of the iterative function during one entire round are those which pass through the value 0.5 in the first round. A convergent cycle must attract at least one critical point; therefore all convergent cycles can be obtained by just shifting the iteration sequence, and keeping the starting value 0.5. In practice, shifting this sequence leads to changes in the fractal, as some branches get covered by others; notice for instance how the Lyapunov fractal for the iteration sequence AB is not perfectly symmetric with respect to a and b.

Newton Fractal:
The Newton fractal is a boundary set in the complex plane which is characterized by Newton’s method applied to a fixed polynomial . When there are no attractive cycles, it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, . In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits a complex appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point.

Many points of the complex plane are associated with one of the roots of the polynomial in the following way: the point is used as starting value z0 for Newton’s iteration , yielding a sequence of points z1, z2, …. If the sequence converges to the root ζk, then z0 was an element of the region Gk. However, for every polynomial of degree at least 2 there are points for which the Newton iteration does not converge to any root: examples are the boundaries of the basins of attraction of the various roots. There are even polynomials for which open sets of starting points fail to converge to any root: a simple example is z3-2z+2, where some points are attracted by the cycle 0, 1, 0, 1 … rather than by a root.

To plot interesting pictures, one may first choose a specified number d of complex points (ζ1,…,ζd) and compute the coëfficients (p1,…,pd) of the polynomial
.

Then for a rectangular lattice zmn = z00 + mΔx + inΔy, m = 0, …, M - 1, n = 0, …, N - 1 of points in , one finds the index k(m,n) of the corresponding root ζk(m,n) and uses this to fill an M×N raster grid by assigning to each point (m,n) a colour fk(m,n). Additionally or alternatively the colours may be dependent on the distance D(m,n), which is defined to be the first value D such that |zD - ζk(m,n)| < ε for some previously fixed small ε > 0.

Nova Fractal:

Nova fractal refers to a family of fractals related to the newton fractal. The formula was named by Paul Derbyshire. Nova is a formula that is implemented in most Fractal Art software. There are a number of related variants of the nova fractal formula, that combine parts of other fractals such as the phoenix fractal and the halley fractal, as well as variants that “double-up” parts of the calculation.

The nova fractal formulae have names like DoubleNova, HalleyNova and PhoenixDoubleNova, each one has a Mandelbrot and a Julia variant.

Burning Ship Fractal:

The Burning Ship fractal, first described and created by Michael Michelitsch and Otto E. Rössler in 1992, is generated by iterating the function:

in the complex c-plane which will either converge or escape. The difference between this calculation and that for the Mandelbrot set is that the real and imaginary components are set to their respective absolute values before squaring at each iteration. The mapping is non-analytic because its real and imaginary parts do not obey the Cauchy-Riemann conditions.

Check out these cool fractals videos to get a better idea:

This one even explains the different styles of fractals while showing them: