Fractal
[1][2][3][4] This exhibition of similar patterns at increasingly smaller scales is 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, the shape is called affine self-similar.
"[11] Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is 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.
[12] The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples have been formulated and studied.
[5][4][13][14][15][16] Fractal patterns with various degrees of self-similarity have been rendered or studied in visual, physical, and aural media[17] and found in nature,[18][19][20][21] technology,[22][23][24][25] art,[26][27] and architecture.
[28] Fractals are of particular relevance in the field of chaos theory because they show up in the geometric depictions of most chaotic processes (typically either as attractors or as boundaries between basins of attraction).
[30] 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.
So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4.
[1]: 405 Indeed, according to various historical accounts, after that point few mathematicians tackled the issues and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters".
[33][8][9] Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable at the Royal Prussian Academy of Sciences.
One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake.
By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what is now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals.
Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings).
[1]: 179 [33][9] That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain?
Statistical Self-Similarity and Fractional Dimension,[37][38] which built on earlier work by Lewis Fry Richardson.
A straight line, for instance, is self-similar but not fractal because it lacks detail, and is easily described in Euclidean language without a need for recursion.
Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space.
Fractal patterns have been reconstructed in physical 3-dimensional space[24]: 10 and virtually, often called "in silico" modeling.
[46] Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above.
[4][13][24] As one illustration, trees, ferns, cells of the nervous system,[21] blood and lung vasculature,[46] and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques.
Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges.
Ian Wong and co-workers have shown that migrating cells can form fractals by clustering and branching.
Diego Krapf has shown that through branching processes the actin filaments in human cells assemble into fractal patterns.
[75] Similarly Matthias Weiss showed that the endoplasmic reticulum displays fractal features.
[77][78][79] Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.
Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, trade, and architecture.
Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns.
