Fractal generally has the following characteristics: [2]
Fine structures can be found on any small scale;
It is so irregular that it is difficult to describe it in the language of traditional Euclidean geometry.
Self-similarity (at least roughly or arbitrarily)
Hausdorff dimension will be greater than topological dimension (except for spatial filling curves such as Hilbert curves);
There is a simple recursive definition.
Because fractals are similar on all scales, they are usually considered infinitely complex (in imprecise terms). Things similar to fractals in nature include clouds, mountains, lightning, coastlines and snowflakes. However, not all self-similar things are fractal. Although the solid line is self-similar in form, it does not conform to other characteristics of fractal.
/kloc-in the 7th century, the mathematician and philosopher Leibniz thought about recursive self-similarity, and fractal mathematics gradually took shape (although he mistakenly thought that only straight lines would be self-similar).
Until 1872, Karl Veiershtrass gave a function that is continuous but differentiable everywhere. Today, it is considered as a fractal graph. 1904, Koch van Kaka was dissatisfied with Weiher's abstract and analytical definition, and gave a definition with similar function but more geometric significance, which is today's Koch snowflake. The following year, the Scherbinsky carpet was made.1904. At first, these geometric fractals were considered as fractals, and ...10086.00000000000606 1938 Paul Pierre Lé vy further put forward the concept of self-similar curve in his paper "Curves and Surfaces in Plane or Space Composed of Parts Similar to the Whole", in which he described a new fractal curve-Levi's C-shaped curve.
Georg Cantor also gave a subset of real numbers with unusual properties-Cantor set, which is also considered as fractal today.
Iterative functions of complex plane were studied by Jules Henri Poincare, Felix Klein, Pierre Fatu and gaston Joulia in the end of 65438+2009 and the beginning of the 20th century, but until now, many functions they found showed their beauty with the help of computer drawing.
In the 1960 s, Benhua Mandelberg began to study self-similarity and wrote a paper, "How long is the coastline of Britain? Statistical self-similarity and fractal dimension. Finally, in 1975, Mandelberg put forward the word "fractal" to mark an object, and its Hausdorff dimension will be greater than the topological dimension. Mandleberg described this mathematical definition with excellent computer architecture images, which are universal; Many of them are based on recursion and even the general meaning of fractal.
Legislation
The four general techniques for making fractals are as follows:
Escape time fractal: it is defined by the recurrence relation of points in space (such as complex plane), such as Mandleberg set, Joulia set, burning ship fractal, new fractal, Leopold fractal, etc. The two-dimensional vector field generated by one or two iterations of the escape time formula will also generate fractal, if the point passes through this vector field repeatedly.
Iterative function system: These fractals have fixed geometric substitution rules. Cantor set, Scherbinsky triangle, Scherbinsky carpet, space filling curve, Koch snowflake, dragon curve, T-square and Munger sponge are all examples of this fractal.
Random fractal: produced by random and uncertain processes, such as Brownian motion trajectory, Levi's flight, fractal landscape, Brownian tree, etc. The latter will produce a kind of fractal called tree fractal, such as diffusion-limited aggregation or reaction-limited aggregation cluster.
Strange attractor: It is generated by a mapping or an iteration of a set of initial differential equations that will show chaos.
[edit] classification
Fractal can also be classified according to its self-similarity, and there are the following three types:
Exact self-similarity: This is the strongest self-similarity, and the fractal looks the same at any scale. Fractal defined by iterative function system usually shows accurate self-similarity.
Semi-self-similarity: this is a loose self-similarity, and the fractal will behave roughly the same (but not accurate) at different scales. Semi-self-similar fractal includes the reduced size of global fractal deformation and degradation. The fractal defined by recursive relation is usually semi-self-similar, not completely self-similar.
Statistical self-similarity: this is the weakest self-similarity, and this fractal can maintain a fixed numerical value or statistical measure on different scales. Most reasonable definitions of "fractal" will naturally lead to some kind of statistical self-similarity (fractal dimension itself is a numerical measure that remains fixed at different scales). Random fractal is an example of statistical self-similarity, but it is inaccurate and semi-self-similar.