Fractal Geometry

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge. Fractals can also be nearly the same at different levels. Fractals also includes the idea of a detailed pattern that repeats itself. Read more ...

Fractals In the News ...

Fractal edges shown to be key to imagery seen in Rorschach inkblots   Science Daily - February 14, 2017

Researchers have unlocked the mystery of why people have seen so many different images in Rorschach inkblots. The image associations are induced by fractal characteristics at the edges of the blots and depend on the scaling parameters of the patterns, says researcher. Fractals are objects with irregular curves or shapes and are recognizable building blocks of nature. Trees, clouds, rivers, galaxies, lungs and neurons are fractals. The new discovery isn't about improving inkblots for psychological assessments -- their use became controversial and mostly set aside in the last 20 years. It does, however, have implications for Taylor's efforts to design a fractal based retinal implant and for potentially improving materials used for camouflage.

Researchers find evidence of fractal behavior in pulsating stars   PhysOrg - February 4, 2015

A team of researchers working at the University of Hawaii using data from the Kepler space telescope, has found that the oscillations made by a star conform closely to the golden mean further study showed that it also behaves in a fractal pattern. In studying the Kepler data, the team was able to track the pulses that emanated from the star over a period of four years taken at 30 minute intervals. They found that two of star KIC 5520878's pulsating frequencies occurred at 4.05 and a 6.41 hour cycles - which the team noted had a ratio of 1.58, which is close to 1.618, aka the Golden Ratio -famously found in nature and sometimes artistic renderings. Intrigued, they looked deeper and found that the frequencies conformed to fractal patterns separating the oscillations into their constituent parts revealed additional weaker frequencies, similar to the way, the team points out, that images of shorelines display craggy lines regardless of how close or far away they are viewed from.

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.

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

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 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

Fractals and Crop Circles