[SuchFriendsAreDangerous]

For the 2009 recording by Skyfire, see Fractal (EP).

 

 

The Mandelbrot set is a famous example of a fractal


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,"[1] a property called self-similarity. Roots of mathematical interest on fractals can be traced back to the late 19th Century, the term however was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2]
A fractal often has the following features:[3]

• 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).[4]
• 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, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. 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 possible to zoom into a region of the image that does not exhibit any fractal properties.


6]

[noisereductions]

http://en.wikipedia.org/wiki/Autechre

[SuchFriendsAreDangerous]

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 curve. (The image at right is three Koch curves put together to form what is commonly 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 recognized a "supreme beauty" within the emerging mathematics of fractals.[2] 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".

[edit] Examples

 

 

A Julia set, a fractal related to the Mandelbrot set


A class of examples is given by the Cantor sets, Sierpinski triangle and carpet, Menger sponge, dragon curve, space-filling curve, and Koch curve. 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 (see attractor). Objects in the parameter space for a family of systems may be fractal as well. An interesting example is the Mandelbrot set. This set contains whole discs, so it has a Hausdorff dimension equal to its topological dimension of 2—but what is truly surprising is that the boundary of the Mandelbrot set also has a Hausdorff dimension of 2 (while the topological dimension of 1), a result proved by Mitsuhiro Shishikura in 1991. A closely related fractal is the Julia set.

[edit] Generating fractals

 

 

[SuchFriendsAreDangerous]

Even 2000 times magnification of the Mandelbrot set uncovers fine detail resembling the full set. Four common techniques for generating fractals are:

• Escape-time fractals – (also known as "orbits" fractals) These are defined by a formula or 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-Highway 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.

[edit] Classification

Fractals can also be classified according to their self-similarity. There are three types of self-similarity found in fractals:

• Exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different scales. Fractals defined by iterated function systems often display exact self-similarity.
• Quasi-self-similarity – This is a loose form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar but not exactly self-similar.
• Statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of "fractal" trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar, but neither exactly nor quasi-self-similar.

[edit] In nature

Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees. This connection is hoped to help determine and solve the environmental issue of carbon emission and control. [5]
In 1999, certain self similar fractal shapes were shown to have a property of "frequency invariance"—the same electromagnetic properties no matter what the frequency—from Maxwell's equations (see fractal antenna).[

[noisereductions]

http://en.wikipedia.org/wiki/Intelligent_Dance_Music

[SONIC GAIL]

How do you keep all this shit in yr head an smoke so much?

[SuchFriendsAreDangerous]

this shit is crazy.

aside from giving us bad ass computer generated images and powerful cell phone antennas, fractal geometry is also involved in practically EVERYTHING in the universe.

Almost every conceivable pattern is based upon fractal geometry. Effects of gravity or small/weak nuclear forces.. cell structures, blood flow.. shit even the trees in the forest grow in fractal patterns to increase their effeciency at carbon dioxide intake (much like the fractal cell phone antenna gets exponentially stronger reception)..

the very universe itself, at every level, is structured around fractals. That shit is trippy beyond their kaleidoscope appearances..
further the mathematical concept of infinity is only truly represented in fractals, and infinity is something hard to conceive in the human mind. I often wonder why folks like Francis Collins do not dive into Fractal Geometry to help support their religious claims with solid mathematical evidence, I find fractal evidence far more convincing of some divine Creator than I do DNA or the Anthropic Principle..

discuss.

[noisereductions]

 

[Rob Instigator]

fractals ar where it's at

[SYRFox]

Fractal is a digital-only EP by Swedish metal band Skyfire. It was made available for digital download on July 6, 2009. The EP contains four songs, which were recorded between the release of Spectral in 2004 and their upcoming album Esoteric to be released in September 2009.[1] The EP features the debut of two new members: guitarist Johan Reinholdz and Joakim Karlsson. The last track "Esoteric" was previewed on their official MySpace page and was made available for download.[1]
Contents

[hide]

• 1 Track listing
• 2 Personnel
  • 2.1 Band members
  • 2.2 Production
• 3 References

//
[edit] Track listing

1. "The Land of the Wolves" – 6:03
2. "Bereaved in Denial" – 4:37
3. "The Transgressor Within" – 6:32
4. "Esoteric" – 4:43

[edit] Personnel


[edit] Band members

• Joakim Karlsson – vocals
• Martin Hanner – bass guitar, keyboards
• Andreas Edlund – guitars, keyboards
• Johan Reinholdz – guitars
• Joakim Jonsson – drums

[edit] Production

• Recorded, mixed and mastered at Powernest Studios with the exception of the track "Esoteric", which was mixed and mastered by Jonas Kjellgren
• Vocals on all tracks produced by guitarist Patrik Persson (The Forsaken)

[edit] References

1. ^ a b "Skyfire: Digital-Only EP Details Revealed". blabbermouth.net. May 19, 2009. http://www.roadrunnerrecords.com/bla...sitemID=120402. Retrieved 2009-07-08.

 

This 2000s death metal album-related article is a stub. You can help Wikipedia by expanding it. Retrieved from "http://en.wikipedia.org/wiki/Fractal_(EP)"
Categories: 2000s death metal album stubs | 2009 EPs | Skyfire albums

[noisereductions]

 

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[floatingslowly]

as above, so below.

[notyourfiend]

Originally Posted by SuchFriendsAreDangerous

For the 2009 recording by Skyfire, see Fractal (EP).

 

 

The Mandelbrot set is a famous example of a fractal


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,"[1] a property called self-similarity. Roots of mathematical interest on fractals can be traced back to the late 19th Century, the term however was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2]
A fractal often has the following features:[3]

• 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).[4]
• 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, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. 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 possible to zoom into a region of the image that does not exhibit any fractal properties.


6]

Once, way back when, I did drugs and saw this stuff everywhere.

[SuchFriendsAreDangerous]

Originally Posted by notyourfiend

Once, way back when, I did drugs and saw this stuff everywhere.

that is the idea! you don't even need to take drugs, just close and squint yr eyes to hard for a second and the kaleidescope patterns you see are fractals. Look into the designs of everything in nature, from flood plains to conch shells. On a fundamental level, the very structure of a third/fourth dimensional universe is based upon fractal geometry and the concept of tesseract.

[notyourfiend]

Originally Posted by SuchFriendsAreDangerous

that is the idea! you don't even need to take drugs, just close and squint yr eyes to hard for a second and the kaleidescope patterns you see are fractals. Look into the designs of everything in nature, from flood plains to conch shells. On a fundamental level, the very structure of a third/fourth dimensional universe is based upon fractal geometry and the concept of tesseract.

It was actually pretty cool, I started thinking in fibernachi sequences.

Wow, I never actually released that those were fractals. I've always been fascinated by those closed eyes sequences. So neat! Fractals rule!

[SuchFriendsAreDangerous]

Originally Posted by SuchFriendsAreDangerous

6]

this shit is everywhere. Think how often we encounter it in our daily routines..

the fractal/spiral is the fundamental structure of the universe, at every level, from the macro to the nano..

 

 

(this is why the northern pacific has become a fucking garbage pile, all the trash spirals into the region from the whole damned world!)

even in political/social organizations, purely man made creations, we use the spiral.

 

[Glice]

I remember the point at which this sort of shit blew my mind. It was precisely the point at which I realised that marijuana was a dangerous and debilitating drug that threatened to take whatever intelligence I had/ have away from me.

[pbradley]

Originally Posted by SuchFriendsAreDangerous

 

This image demonstrates that the nautilus shell isn't a golden spiral.

[pbradley]

 

Bitches don't know about my rationalism.

[phoenix]

math is indeed awesome. I wish I could go back to school/uni and actually take math subjects, instead of just sitting in for fun.