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This nOde last updated December 17th, 2004 and is permanently morphing...
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A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
[French, from Latin frâctus, past participle of frangere, to break.]
Fractal, any of many geometric shapes that are complex and detailed at any scale. Fractals are often self-similar- that is, each portion is a reduced-scale replica of the whole. Many such self-repeating figures can be constructed.
French mathematician Benoit B. Mandelbrot discovered fractal geometry in the 1970s. Mandelbrot adopted an abstract definition of dimension, with the result that a fractal cannot be treated mathematically as existing in one, two, or any other whole-number dimensions. It must be treated as having some fractional dimension.
A coastline, if measured at progressively smaller scales, would tend toward infinite length. Mandelbrot has suggested that mountains, clouds, galaxy clusters, and other natural phenomena are similarly fractal in nature. The beauty of fractals has made them a key element in computer graphics. Fractals have also been used to compress still and video images in a variety of multimedia and other image-based computer applications.
fractal (frak'tel) noun
A word coined by mathematician Benoit Mandelbrot in 1975 to describe a class of shapes characterized by irregularity, but in a way that evokes a pattern. Computer graphics technicians often use fractals to generate naturelike images such as landscapes, clouds, and forests. The distinguishing characteristic of fractals is that they are "self-similar"; any piece of a fractal, when magnified, has the same character as the whole. The standard analogy is that of a coastline, which has a similar structure whether one is looking at the coastline of a state or that of a continent. Interestingly, it is often difficult to measure the length of the perimeter of such a shape exactly because the total distance measured depends on the size of the smallest element measured. For example, one could measure on a given coastline the perimeter of every peninsula and inlet, or at a higher magnification the perimeter of every small promontory and jetty, and so on. In fact, a given fractal may have a finite area but an infinite perimeter; such shapes are considered to have a fractional dimension- for example, between 1 (a line) and 2 (a plane)- hence the name fractal.
There are points in _Writing On
Drugs_ where Sadie
Plant flirts with the idea that drugs can access certain "revelations."
is that it's not a transcendent reality
"out there," but one deep within the hard wiring
of the brain itself. She subscribes to Henri Michaux's mescaline-inspired conviction
that there's a kind of pre-cultural commonality underlying all the many forms
experience through history and across the globe. The deranged geometry of lattices, honeycombs,
lacework, and spiderwebbing, the baroquely infolding spirals and proliferating
ornamentation, and the mosaic
vision and kaleidoscopic turbulence, seen by users of LSD, peyote, DMT, psilocybin,
and other hallucinogens,
find a visual echo in such cultural forms as the "coptic light"
patterns of Arabian carpets and the paisley fabric of the Indian subcontinent.
Michaux speculated that all this drug-induced eye candy constitutes an amplification
of brain wave activity, especially that of the visual cortex. The fact
that some migraine sufferers see similar patterns -- known as the migraine
aura -- suggests that in certain extreme states, the MS/DOS
and subroutines of the brain can be apprehended by consciousness. "Some people
can get the aura effects without the pain of migraine," says Plant. "It's
happened to me about three times in my life, at times of extreme exhaustion.
This almost kaleidoscopic stuff kind of creeps across your visual field from
one side to the other. It's really quite stunning, and not at all scary.
The fact that there are 'natural' equivalents to drug-induced experiences suggests
the possibility you are in some sense observing what's going on in the brain."
Noting the similarity between these psychedelic hallucinations and the self-similar
patterns of Mandelbrot's fractals, Plant characterizes the drugged or migrained
brain as a cranked-up biochemical computer capable of picturing the self-organizing
behavior and nonlinear dynamism at play within normally staid reality.
So I think about all of this all
the time, and I feel great change. I try to monitor it, especially in the realm
of society and technology. Everything is redefined every 30 days, every
60 days, redefined toward some kind of singularity,
some kind of extra-ordinary moment in the fractal pattern of Historical unfoldment.
You know, fractals are always repetitious, always low levels build to higher
levels, but nevertheless, intrinsically to the pattern, there comes a moment
where there is an apotheosis, a breakthrough to a new level of understanding.
And then whatever the old world was, it simply dissipates. It goes away.
Not that there isn't political struggle, but once the (let's call it) karmic
underpinnings of a historical position -- especially an oppressive historical
position -- once those underpinnings are articulated, revealed, shown in the light
of day, then the game cannot continue.
-Terence McKenna - _Live at Wetlands Preserve, NYC July 28, 1998_
Devised by Benoit Mandelbrot from the adjective latin
term Fractus, from the verb Frangere, to break. Theresonance
of the main English cognates - Fracture and Fraction (noun and adjective (English
and French)). Mandelbrot devised the word Fractal to describe a new mathematical
geometry, part of the new science of chaos.
"A fractal is a geometric shape, a geometric shape having a special property
that as you look closer and closer to it you see it is essentially the same
thing." The notion of self-similarity strikes ancient chords in our culture.
An old strain in western thought honors the idea. Leibniz
imagined that a drop of water
contained a whole teeming universe containing in turn waterdrops and new universes
within. "To see the world in a grain of sand." - Blake. When spermatozoa
was first discovered it was thought to be an homunculus, a human, tiny, but
fully formed. But self-similarity receded as a scientific principle, and
of ontogenetic development is far more interesting than mere enlargement.
The myth died hard as the human vision was extended by telescopes and microscopes.
The first discoveries were realizations taht each change of scale brought new
phenomena and new kinds of behavior. For modern particle physicists, the
process has never eneded. Every new accelerator with its increase in energy
and speed, extends sciences field of view to tinier particles and briefer time
scales and every extension seems to bring new information.
But physicists wanted to know more. They wanted to know why. There
were forms in nature, not visible forms, but shapes embedded in the fabric of
motion, waiting to be revealed.
- from the liner notes of track _Fractalize_ by ClockDVA off of _Man-Amplified_ CD on Contempo (1992)
The final crop
circle formation of 1997, was the "Strange
Attractor" fractal at Hackpen Hill . The term means "an irreducible
invariant set that attracts the trajectories of all nearby points." This term
was used by Terence
McKenna to describe the "Omega
Point," which sucks our evolutionary
trajectory towards it, like the plughole at the end of time.
In recent years, chaos has migrated from mathematics, physics, and computer science departments to fields as far off as anthropology and economics, displaying a viral panache similar to poststructuralism's as it spread through the humanities. And perhaps it's no mere coincidence. In "Non-Organic Life," an article in Zone Incorporations, Manuel DeLanda persuasively links the chaotic process of self-organization with Deleuzian poststructuralism, suggesting a more-than-metaphoric link between natural and social formations and sketching out a potent ethics offlow. Even literary theory is now surfing the fractal edge.
- Erik Davis - _Lost In Hyperspace_
WHEN COSMOLOGISTS SPECULATE on the nature of the Universe, they make an unspoken assumption--that matter is spread uniformly throughout space. Yet when astronomers peer out across the Universe they see something very different.
Galaxies are gathered together in great chains and walls which snake around vast regions of empty space called voids.
The Universe appears anything but uniform.
"Yes, there does appear to be a contradiction," admits Ofer Lahav of the Hebrew University in Jerusalem and the Institute of Astronomy in Cambridge. But Lahav contends that the Universe, though undeniably clumpy on the small scale, becomes smooth on the largest scales. "I like to think of it as an ocean, which looks choppy on the scale of individual waves but from far above, on scale of tens of kilometres, is perfectly smooth," he says.
A maverick group based in Europe has suggested that the Universe never becomes smoothed out, even on the largest scales. "My contention is that it is clumpy on all the scales so far explored," says Francesco Sylos Labini, an astronomer at the University of Geneva. "In fact, studies we have done show that the distribution of matter is fractal, just like a tree or a cloud."
If this dissenting view is correct and the Universe doesn't become smoothed out on the very largest scales, the consequences for cosmology are profound. "We're lost," says Coles. "The foundations of the big bang models would crumble away.
We'd be left with no explanation for the big bang, or galaxy formation, or the distribution of galaxies in the Universe."
"Our tests show that the Universe never becomes homogeneous in the available galaxy samples," says Sylos Labini, who began his work while in Pietronero's team.
"It remains hierarchically clustered. It remains fractal."
"If the Universe is fractal, however, it has no characteristic scale," says Sylos Labini. "Everything, including the average density, changes with scale so the concept is meaningless.
It's not surprising that people find the Universe is homogeneous when homogeneity is one of their basic assumptions."
Despite this formidable theoretical challenges, Sylos Labini believes a fractal Universe is more exciting than a homogeneous one. "We are facing a new and challenging problem which we are a long way from solving," he says. "For some questions, the fractal structure leads to a radically new perspective and this is hard to accept. But it is based on the best data and analyses available. It is neither a conjecture nor a model--it is a fact."