Fractal Geometry

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 Benoit Mandelbrot in 1975 and was derived from the Latin fractious meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.

A fractal often has the following features:

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; for instance, it is regular enough to be described in Euclidean terms.

Images of fractals can be created using fractal generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, as it is quite possible to zoom into a region of the image that does not exhibit any fractal properties.

History of Fractals

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.

In 1918, Bertrand Russell had recognized a "supreme beauty" within the mathematics of fractals that was then emerging. The idea of self-similar curves was taken further by Paul Pierre Levy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Levy 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 Poincare, 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, Benoit 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".

Classic Examples of Fractalization

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. Benoit Mandelbrot and others worked hard to communicate this area of mathematics to the public.

'Fractal' mathematician Benoit Mandelbrot dies aged 85   BBC - October 18, 2010

Benoit Mandelbrot, who discovered mathematical shapes known as fractals, has died of cancer at the age of 85. Mandelbrot, who had joint French and US nationality, developed fractals as a mathematical way of understanding the infinite complexity of nature. The concept has been used to measure coastlines, clouds and other natural phenomena and had far-reaching effects in physics, biology and astronomy.

The quest for Mandelbrot fractals in 3D   New Scientist - November 18, 2009

The Koch Snowflake

To create a Koch snowflake, one begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral "bump." One then performs the same replacement on every line segment of the resulting shape, ad infinitum. With every iteration, the perimeter of this shape increases by one third of the previous length. The Koch snowflake is the result of an infinite number of these iterations, and has an infinite length, while its area remains finite. For this reason, the Koch snowflake and similar constructions were sometimes called "monster curves."

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. Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set.

The Julia Set

Some of the most remarkable fractals are the Julia sets, devised by the French mathematician Gaston Julia (1893-1978). The Julia Sets are generated by applying an iterative non-linear process based on a very simple square-law function.

F(z,C) = z2 + C

where z is a point on the X-Y plane and C is a constant with both x and y components, Cx and Cy. The results were very surprising. No one expected that such a simple function could produce such complex images and be so difficult to analyze.

Cantor Sets

Sierpinski Triangle

Menger Sponge

Dragon Curve

Space-filling Curve

Lyapunov fractal

Fractal Dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension and none of them should be treated as the universal one. From the theoretical point of view the most important are the Hausdorff dimension, the packing dimension and, more generally, the Renyi dimensions. On the other hand the box-counting dimension and correlation dimension are widely used in practice, partly due to their ease of implementation.

Although for some classical fractals all these dimensions do coincide, in general they are not equivalent. For example, what is the dimension of the Koch snowflake? It has topological dimension one, but it is by no means a curve - the length of the curve between any two points on it is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. In some sense, we could say that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the question of whether its dimension might best be described in some sense by number between one and two. This is just one simple way of motivating the idea of fractal dimension.

Fractals and Crop Circles

  Arthur Clarke - Explaining Fractals - The Colors Of Infinity 1 of 6 YouTube

Fractals In the News ...

Fractal patterns of early life revealed

New Scientist - July 15, 2004

Newly uncovered fossils reveal in extraordinary clarity the strangeness of the Earth's earliest complex life. The finds show that the organisms were assembled in fractal patterns from frond-like building blocks. They were unable to move and had no reproductive organs, perhaps reproducing by dropping off new fronds. The creatures, which were neither animals or plants, are called "rangeomorphs". They first appeared on the ocean floor 575 million years ago, after the last global glaciation, and were among the first of the soft-bodied creatures in the Ediacaran period. This biota survived until 542 million years ago, when modern animals diversified rapidly in the Cambrian explosion and most Ediacaran species vanished.

Until now, almost all Ediacaran fossils were squashed flat, and the few that were not were poorly preserved. This led to debate over whether the poor preservation obscured links to later life, or if the Ediacaran organisms were in fact a failed experiment in evolution that simply became extinct. The newly unearthed fossils, from Newfoundland, Canada, were preserved three-dimensionally in fine-grained mud by a "one-in-a-million" streak of luck, says Guy Narbonne of Queen's University in Kingston, Ontario.

Volcanic eruption

Just after a mud flow entombed the organisms, a nearby volcano erupted, covering it with a thick ash protective ash deposit. Later, the bed escaped the strain that altered most of the rock in the region. Now weathering is exposing them so "they basically pop out of the rock," Narbonne told New Scientist. "You are seeing what they looked like when they were alive." That exceptional preservation is cracking the mystery of Ediacara. In some spots the surface has eroded and "we see for the first time what was inside an Ediacaran fossil," Narbonne says. Each frond element, a few centimetres long, was made of many tubes held up by a semi-rigid organic skeleton.

The frond elements had branches which themselves had branches, a classic fractal structure. Frondlets assembled themselves like building blocks to make larger living structures attached to the sea floor. Narbonne found rangeomorphs assembled in several different shapes, which he believes filtered food from different levels of the water column, as well as isolated free-living frondlets.

The fractal patterns look complex, but Narbonne says their self-similarity means that very simple genomes - expected in early organisms - would suffice both to assemble individual frondlets and to control their assembly into larger structures. That would explain why the rangeomorphs evolved first. They accounted for over 80% of fossils early in the Ediacara period, when there were no mobile animals or traces of burrows. But they declined as more mobile animals evolved, apparently unable to compete, or perhaps being eaten themselves.

Frond and Symbology Crystalinks