Fractals Definition and 62 Threads

In mathematics, a fractal is a subset of Euclidean space with a fractal dimension that strictly exceeds its topological dimension. Fractals appear the same at different scales, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly smaller scales, a property called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar. Fractal geometry lies within the mathematical branch of measure theory.
One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.Analytically, most fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still topologically 1-dimensional, its fractal dimension indicates that it also resembles a surface.

Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".The consensus among mathematicians is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures, and sounds and found in nature, technology, art, architecture and law. Fractals are of particular relevance in the field of chaos theory because the graphs of most chaotic processes are fractals. Many real and model networks have been found to have fractal features such as self similarity.

View More On Wikipedia.org
  1. C

    Explore Fractals at the New Fractal Forum!

    Are any of you interested in fractals either as science or art? I am trying to open a new forum about fractals in order for peopleto share fractal images, formulas, theoretical backround, info etc. This is brand new so be one of the first people to ever post in http://www.fractalforum.tk" .
  2. H

    Looking for quotes on sequences, and fractals

    Hello, I'm looking for some quotes about sequences, fractals and chaos. Any kind of help is welcome. Thanks :)
  3. H

    Tricks and Pranks in sequences and fractals

    Hello, It's rather a formal question, not asking for anything specific. I'm writing a pseudo-book for my math class, which should contain 1. arithemtic sequences 2. geometric sequences 3. fractals. And so here's my ask for help and a question, Do you know anything about any of these...
  4. A

    Fractals, Chaos and Non-linear Dynamics

    in the movie "the bank" a mathematical genius predicts the exact movements of the sharemarket after years of research and attempts. he uses Fractal geometry, chaos theory, non-linear dynamics and of special interest to him was the work of mandelbrot and his work regarding fractals. He...
  5. D

    How Are Fractals Models of CHAOS?

    Look around you. Everyday objects, linked together. How? By mathematics. Not a simple design, but a complex wonderful array of chaos models called fractals. These "fractals" are ever changing, always on the brink of a different state. Does this sound like math? Believe it or not, all of the...
  6. I

    Why do fractals and Pi have a special relationship?

    This was brought to my attention today, and I haven't had much time to think about it; I think it has something to do with fractals. If you have half a circle with diameter of 2, the circumference will be \pi. If you create two circles, each with diameter of 1, the combined length of the...
  7. Loren Booda

    Can fractals sum to a linear function?

    Does there exist a set of fractals whose sum defines a differentiable field?
  8. S

    Can Zeta-Function Determine All Fractals or Are Beach Photos Better?

    Does the Zeta-function provide every fractal their is or should i take beach photoes? :wink:
  9. M

    Fractals on Google: Why the Logo Makeover?

    www.google.com , they play around with the logo every once in a while but why are they dressing it up with fractals?
  10. D

    Practical applications of fractals?

    What are the practial appliactions of say, fractals? I suppose in some sense they describe objects that appear in nature, eg. trees. But what the use of say Mandelbrot. It just complicates things...
  11. Loren Booda

    Cardinal number: irrationals vs fractals

    How does the cardinal number for the set of irrational numbers compare to that for a fractal set?
  12. Loren Booda

    Fractals of rational dimension and fractals of integral powers

    What generalizations can be made concerning fractals of nonzero rational dimensions M/N (where M and N are nonzero integers)? How does a fractal of non-integral dimension F compare geometrically to a fractal of dimension GF, where G is a nonzero integer?
Back
Top