Hmm,...what a beautiful topic???
till mr.ikhsan said to Lia, "i deal with this title and i'll guide it!" without think a second, ogh,.!
it's different when i show mine,..
vector differentiation of square form !
mr. ikhsan ask me to explain--again and again--before deal, need long argument and sweat to fight it. finally, he said okey,..and you'll be guided by mrs. erni !
fiuhhh,..Alhamdulillah,..!
i hav headache again and again when looked for the title for my colloquium class.
ok,.today i'll share u some about fractals, check it out!
Fractals
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," 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.
A fractal often has the following features:
* 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).
* 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, even various vegetables (cauliflower and broccoli).
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.
Generating fractals
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.
* 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.
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.
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.
Hmm,...i think it's hard to prove in math formula.
ok,..that's all for this shine monday,.many thanks to wikipedia !
see ya,..^o^ !
