Fractal Geometry
Reality is based on the patterns of
sacred geometry which repeat in endless cycles.
Arthur Clarke - Explaining Fractals - The Colors Of Infinity 1 of 6 YouTube
Reality is based on the patterns of
sacred geometry which repeat in endless cycles.
Arthur Clarke - Explaining Fractals - The Colors Of Infinity 1 of 6 YouTube
A fractal is a geometric object which is rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin fractus or "broken".
Before Mandelbrot coined his term, the common name for such structures (the Koch snowflake, for example) was monster curve. Fractals of many kinds were originally studied as mathematical objects. Fractal geometry is the branch of mathematics which studies the properties and behavior of fractals. It describes many situations which cannot be explained easily by classical geometry, and has often been applied in science, technology, and computer-generated art. The conceptual roots of fractals can be traced to attempts to measure the size of objects for which traditional definitions based on Euclidean geometry or calculus fail.
Fractals Wikipedia
The basic unit of the Koch snowflake, first constructed by the mathematician Helge von Koch (1870-1924), is the equilatorial triangle which can be built up into a much larger but still similar pattern. Any part of the snowflake is equally crinkly, whatever scale it is viewed at.
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.
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.
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.
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 Rényi 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.
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
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