Introduzione alla modellazione di forme naturali tramite geometria frattale Corso di Grafica Facoltà di Informatica Magistrale. In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals are encountered. L’arte frattale è creata calcolando funzioni matematiche frattali e trasformando i risultati dei calcoli in immagini, animazioni, musica, o altre forme di espressione.

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In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals are encountered ubiquitously in nature due to their tendency to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot geometdia. One way that fractals are different from finite geometric figures is the way in which 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 grometria 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. As mathematical equations, fractals are usually nowhere differentiable.

The mathematical roots of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century with notions of recursionthen moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard BolzanoBernhard Riemannand Karl Weierstrass[9] 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.

There is some disagreement amongst mathematicians about how the concept of a fractal should be formally defined.

Mandelbrot himself summarized it as “beautiful, damn hard, increasingly useful. The consensus is that theoretical fractals are infinitely self-similar, iteratedand detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. The word “fractal” often has different connotations for laymen as opposed to mathematicians, where the layman is more likely geometdia be familiar with fractal art than the mathematical concept.

The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with little mathematical gekmetria. The feature of “self-similarity”, for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats geometgia and over, or for some fractals, nearly the same pattern geoemtria over and over.

The difference for fractals is that the pattern reproduced must be detailed. This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived.

Now, consider the Koch curve. This number is what mathematicians call the fractal dimension of the Koch curve. The geomehria that the Koch curve has a non-integer fractal dimension is what makes it a fractal.

This also leads to understanding a third feature, that fractals as mathematical equations are “nowhere differentiable “. In a concrete sense, this means fractals cannot be measured in traditional ways. But in geometgia an infinitely “wiggly” fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.


The result is that one must need infinite tape to perfectly cover the entire curve, i. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the rrattale. Bytwo French mathematicians, Pierre Fatou and Gaston Juliathough working independently, arrived essentially simultaneously at results describing what are now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors i.

Different researchers have postulated that without the aid of modern computer graphics, early drattale were limited to what they could depict in manual drawings, so lacked rrattale 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.

In [12] Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word “fractal” and illustrated his mathematical definition with striking computer-constructed visualizations.

These images, such as of his canonical Mandelbrot setcaptured the popular imagination; geomerria of them were based frattald recursion, leading to geometriz popular meaning of the term “fractal”. One often cited description that Mandelbrot published to describe geometric fractals 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”; [1] this is generally helpful but limited.

Authors disagree on the exact definition of fractalbut most usually elaborate on the basic ideas of self-similarity and an unusual relationship with the space a fractal is embedded in. According to Falconer, rather than being strictly defined, fractals should, in addition to being geoetria differentiable and able to have a fractal dimensionbe generally characterized by a gestalt of the following features; [2].

As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, gemetria easily described in Euclidean language, has the same Hausdorff dimension as topological dimensionand is fully defined without a need for recursion.

Images of fractals can be created by fractal generating programs. Because of the butterfly effecta small change in a single variable can frattalle an unpredictable outcome.

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. Models may simulate theoretical fractals or natural phenomena with fractal features.

The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Some specific applications of fractals to technology are listed elsewhere. Images and other outputs of modelling are normally referred to as being “fractals” even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties.

Also, these may include calculation or display artifacts which are not characteristics of true fractals. Modeled fractals may be sounds, [21] digital images, electrochemical patterns, circadian rhythms[50] etc. Fractal patterns have been reconstructed in physical 3-dimensional space [29]: A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.


Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, geometri instance, is currently being used to determine how much carbon is contained in trees. Fractal basin boundary in a geometrical optical frattal [55]. A fractal is formed when pulling apart two glue-covered acrylic sheets. Romanesco broccolishowing self-similar form approximating a natural fractal.

Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO 2.

Width of image is about a kilometer. Slime mold Brefeldia maxima growing fractally on wood. Decalcomaniaa technique used by artists such as Max Ernstcan produce fractal-like patterns. Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are rfattale in African artgames, divinationtrade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on.

Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles. In a interview with Michael SilverblattDavid Foster Wallace admitted that the structure of the first draft of Infinite Jest he gave to his yeometria Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle a. Sierpinski gasketbut that the edited novel is “more like a lopsided Sierpinsky Gasket”.

Humans appear to be especially well-adapted to processing fractal patterns with D values between 1. When two-dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. A fractal in three-dimensional space is similar; such a fractal may have an infinite surface area, but never exceed a certain volume.

If done correctly, the efficiency of the emission process can be maximized. From Wikipedia, the free encyclopedia. For other uses, see Fractal disambiguation.

Self-similarity illustrated by image enlargements. This panel, no magnification. The same fractal as above, magnified 6-fold.

Same patterns reappear, making the exact scale being examined difficult to determine. The same fractal as above, magnified fold, where the Mandelbrot set fine detail resembles the detail at low magnification. Actin cytoskeleton [52] Algae Animal coloration patterns Blood vessels and pulmonary geomwtria [47] Coastlines Craters Crystals [53] DNA Earthquakes [30] [54] Fault lines Geometrical optics [55] Heart rates [22] Heart sounds [23] Lightning bolts Mountain goat horns Mountain ranges Ocean waves [56] Pineapple Psychological subjective perception [57] Proteins [58] Rings of Saturn [59] [60] Gekmetria networks Romanesco broccoli Snowflakes [61] Soil pores [62] Surfaces in turbulent flows [63] [64] Trees Brownian motion generated by a one-dimensional Wiener Process.

Frost crystals occurring naturally on cold glass form fractal patterns. Fractal art and Mathematics and art. The topological dimension and Hausdorff dimension of the image of the Hilbert map geometri R 2 are both 2. Note, however, that the topological dimension of the graph of the Hilbert map heometria set in R 3 is 1. The fractal geometry of nature.

Fractal – Wikipedia

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Physics and fractal structures. MacTutor History of Mathematics.

Arte frattale – Wikipedia

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Arte frattale

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