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Loading ... Fractal universe
The universe is made of successive levels of organizations
and nested like Russian dolls.
Like a giant fractal where each zoom on a detail
reveals new structures, new worlds.
So we will zoom into the world of matter,
of the infinitely large to the infinitely small,
to the limits of current scientific knowledge.
Fractals
Called fractal or fractal (noun least used), a curve or surface shape irregular or fragmented, which is created by following the rules involving deterministic or stochastic internal homothety. The term "fractal" is a neologism coined by Benoît Mandelbrot in 1974 from the Latin root fractus, which means broken, irregular (fractal nf). This term was originally an adjective: the fractal objects. In The Paradoxes of the Treasury (Philip Baker & Alain Cohen, Ed. Belin, 2007), fractals are defined paradoxically, in reference to the nested structures they are special, "The fractal objects can be viewed as nested structures at any point and not just a number of points, attractors of the classic pull-out structure. This design hologigogne (pull-out at any point) of this definition involves tautological fractal: a fractal object is an object in which each element is a fractal object. " Despite appearances, this type of recursive definitions of nature is not only theoretical but also may involve concepts common: an ancestor is a parent or ancestor of a parent, a multiple is a composite of a number or a multiple of that number, a staircase begins or extends a staircase, or a dynasty inaugurated a dynasty extends, etc..
Characteristic
A fractal object has at least one of the following characteristics:
* It has similar details to arbitrarily large or small scales;
* It is too irregular to be described effectively in terms of traditional geometric;
* It is exactly or statistically self-similar, that is to say that the whole is similar to one of its parts;
* Its Hausdorff dimension is strictly greater than its topological dimension. To express it differently, an irrigation system is an unfolding of lines ("1D") that offers features to talk about starting a surface ("2D"). The surface of the lung ("2D") is folded into a kind of volume ("3D"). Pictorially, fractals are characterized by a kind of non-integer dimension.
Domains of validity
Fractals are not required to satisfy all the properties listed above to serve as models. All they need to make appropriate approximations of what interests in a given domain of validity (founder of the book Mandelbrot fractal objects gives a wide variety of examples). The size of the cells of the lung, for example, size at which it ceases to be subdivided so fractal is related to the size of the mean free path of the molecule of oxygen to body temperature.
The size used is Hausdorff, and there it is a new feature uneven surfaces. We know the ranges of validity of the Hausdorff dimension observed on Earth for the mountains, clouds, etc..
Examples of fractals are the Julia sets and Mandelbrot, Lyapunov fractal, Cantor set, the Sierpinski carpet, Sierpinski triangle, Peano curve and the Koch snowflake. Fractals can be deterministic or stochastic fractals. They often appear in the study of chaotic systems.
Fractals can be divided into three broad categories:
1. The iterated function systems. These have a fixed geometric replacement rule (the Cantor set, the Sierpinski carpet, Sierpinski triangle, Peano curve, the Koch snowflake);
2. Fractals defined by a recurrence relation at each point in space (such as the complex plane). Examples of this type are the Mandelbrot set fractal and Lyapunov;
3. Random fractals, generated by stochastic processes, not determine, for example fractal landscapes.
Of all these fractals, only those built by iterated function systems usually display the property of self-similarity, meaning that their complexity is invariant under change of scale.
Random fractals are the most used in practice and can be used to describe many highly irregular objects in the real world. Examples include clouds, mountains, turbulence of liquid, the coast lines and trees. Fractal techniques have also been used in fractal image compression, as well as in many scientific disciplines.
Fractal dimension
The size of a straight line, circle and a smooth curve is 1. Once a fixed origin and meaning, each point on the curve can be determined by a number, which defines the distance between the origin and the point. The negative number is taken whether to move in the opposite direction to that chosen initially.
The size of a single figure in the plane is 2. Once a marker set, each point of the figure can be determined by two numbers. The size of a single body in space is 3.
A figure such as a fractal is not simple. Its size is not as easy to define and is not necessarily right. The fractal dimension, more complex, expressed in terms of Hausdorff dimension.
When the fractal consists of replications of itself smaller, its fractal dimension can be calculated as follows:
d = \ frac {\ ln (n)} {\ ln (h)}
where the initial fractal consists of n copies whose size was reduced by a factor h (dilation).
A few examples:
* One side of the Koch snowflake is formed from n = 4 copies of itself reduced by a factor h = 3. Its fractal dimension is:
d = \ frac {\ ln (4)} {\ ln (3)} \ simeq 1.2618595 ...
* The Sierpinski triangle is formed from n = 3 copies of itself reduced by a factor h = 2. Its fractal dimension is:
d = \ frac {\ ln (3)} {\ ln (2)} \ simeq 1.5849625 ...
* The Sierpinski carpet is composed of n = 8 copies of itself reduced by a factor h = 3. Its fractal dimension is:
d = \ frac {\ ln (8)} {\ ln (3)} \ simeq 1.892789 ...
Fractals in Nature
Approximate fractals are easily observable in nature. These objects have a self-similar structure over a wide range, but finite clouds, snowflakes, mountains, river networks, cauliflower or broccoli, and blood vessels.
Trees and ferns are fractal in nature and can be modeled by computer using a recursive algorithm. The recursive nature is obvious in these examples, the branch of a tree or a fern frond are miniature replicas of the whole: not identical but similar in nature.
The surface of a mountain can be modeled on a computer using a fractal: take a triangle in three dimensional space which we connect the circles on each side by segments, resulting in four triangles. The central points are then moved randomly up or down, in a defined radius. The procedure is repeated, reducing the radius by half at each iteration. The recursive nature of the algorithm ensures that the whole is statistically similar to each detail.
Finally, some astrophysicists have noticed similarities in the distribution of matter in the Universe at six different scales. The collapse of interstellar clouds successive, due to gravity, are the cause of this structure (partial) fractal. This view gave rise to the fractal model of the universe, describing a world based on fractals.
Fractal geometry to fractal art
The science of "fractals", mathematical objects specific to the non-Euclidean geometry, invented by the mathematician Benoit Mandelbrot in the 1960s, has been honored in the scientific literature in 1975 in his seminal book, The fractal object - Form, chance and dimension. This geometry applies to irregular forms of the complex as well as figures of pure mathematics, served as a basis for reflection and creative artists fractalist international movement since the 1980s, regardless of the particular area of their respective investigations arts (visual arts, digital arts, photography, music or literature).
The artistic movement fractalist includes the many creations, extremely varied, artists of different nationalities - Europeans, Japanese, Americans - who founded their creative activity of the reference to the physico-mathematical theory of stochastic complexity (ie ie random) dynamical systems. However, the theory of dynamical systems, which may hold a capacity "of self-organization", it was build substantially in the international scientific community during the 1970s. For scientific discourse, the notion of complex stochastic (or random) implies the idea of indeterministic dynamical processes, not describable by the ordinary laws of mathematical continuity, and therefore unpredictable in the long run. This inability to predict their long-term behavior is because they are able to reorganize itself indefinitely in a new way over time, although some self-organizing systems are, in some cases, almost predictable ( cyclic sequence, but overall predictable trajectory approximation, etc.).. For this reason, they are supposed to be governed "objectively" (really) by the laws of chance. Biologists, meteorologists, sociologists, economists, physicists, chemists, and of course, mathematicians, frequently resort to "laws of chance" to try to understand the complexity of approximate unpredictable phenomena they study.
In summary, a complex stochastic (or random) process actually involves indeterministic, inherently governed by the laws of chance. The great contribution of complexity theory has been to reveal the existence of phenomena simultaneously deterministic and unpredictable. It should be noted finally that the fractal mathematics, in this complex and unpredictable, represent only one aspect of the laws of chance, not the only form they may adopt.
Correspondingly, the artists fractalists admit, at least implicitly, a conceptual model for governing the aesthetic philosophy of creative enterprise, the building mathematical fractal geometry, formalized by the mathematician Benoit Mandelbrot-computer in the years 1960-1970. Fractal geometry can precisely characterize quantitatively some geometric properties specific to the formal representation of dynamic systems. The term "fractal", used as a noun or as adjective, is therefore strictly scientific origin, since it belongs to the vocabulary of the geometry of natural phenomena - microscopic or macroscopic - infinitely irregular and unpredictable in their details at any scale of observation. The language of contemporary geometry thus denominates "fractal object" (or more briefly "fractal") a spatial configuration in non-integer dimension, which "extends" into a full-dimensional space immediately above, and this space can have n ' any number of dimensions (1, 2, 3, or any n). This configuration discontinuous, is apparently very orderly and symmetrical scaling (the curve or von Koch snowflake, which is self-similar at any scale, for example) or very irregular and asymmetrical as the coastline of a coastline or the outlines of a cloud can be characterized, regardless of the level of review used by a variable degree of statistical irregularities that ordinary Euclidean geometry can be measured and it is not known to account satisfactorily. This means that a fractal object can be extremely irregular, but it is not a necessary condition to make a fractal object, while the geometric measure that accounts for it always depends on the scale of examination adopted.
The original art practices claiming a membership of fractal aesthetics is therefore the mathematical study of irregular shapes infinitely in every detail, broken, broken and fragmented in each of their plots, essentially discontinuous (the past participle of Latin "fractus" summarizes these meanings that converge towards the idea to grind, grind and break). The neologism "fractal", created by the mathematician Benoit Mandelbrot in the first French edition of his famous book: The fractal object - Form, chance and dimension (1st ed. 1975, 4th ed. Journal, 1995, Paris, Flammarion), included also the abandonment of traditional mathematical concept of spatial symmetry, associated with Euclidean geometry, in favor of another type of organization governing a complex way the elements of an irregular spatial pattern in all its components. This new "order fractal" was defined in strictly algebraic mathematics as an index of irregularity morphology: the fractal dimension, absolute numbers not designated a measure of size but a measure of the complexity of formal flat or three-dimensional configurations.
What forms can be regarded as irregular and discontinuous infinitely? The examples from nature are everywhere and they are discovering physical expansion continued. The structure of the clouds in motion, the shape of the mountains, the organization of a starry sky, the infinite universe of galaxies, as a simple chestnut leaf, a piece of rock, a piece of metal or a biological cell, human, animal or plant are affected numerous areas of irregularity in terms of levels of observation which they are subjected. The merit of fractal geometry is precisely to be used to characterize these degrees or levels of irregularity morphostructural who sign the heterogeneity of the material and the entire universe.
These are the scales of examination of the object, natural or geometric variables that define the degree of discontinuity. The theme is well known in theoretical physics (quantum mechanics), the procedure of inter-observer with the observed object, this area is establishing itself as the mathematical basis essential to the determination of the fractal dimension. There is an analogy between the positions of the observers in fractal geometry and quantum mechanics: in both cases, the presence of the observer changes the outcome of the experiment in progress. However, to maintain the accuracy of reasoning, it should be noted that the analogy ends there: in quantum mechanics is the very presence of the observer and his measuring instruments, as part of physical reality, which is the disturbance factor. For fractal objects, the result is modified according to the view that the observer chooses to take: the mesoscopic scale level to which it stops. The two situations are, therefore, very different, and therefore lead to two very different models of the world.
From the perspective of fractal objects of nature, seen at distance, as a whole may appear simple shapes, regular, describable by means of the categories of traditional Euclidean geometry: circles, triangles, parallelepipeds, spheres, cones, cylinders, polyhedra, and any combination of these basic primitives. However, observed more closely, these natural forms become more complicated, less linear, less "Euclidean" and they have broken contours and surface structures of branched entangled. If the level of observation, ever more demanding, continues to be refined through a magnifying glass and microscope, the detail appears as a myriad of finer details and richer microform, themselves saturated to infinity microform nested hyper-detailed new appearances.
Of course, the fractal in nature does "not go" to infinity. There is a level of scale limit the nature of this aspect fractal: it turns off when the self-similarity stops. For an object like a rock, it ceases when moving at a molecular level, with no formal self-similarity with the rock itself. The term "fractal" can not therefore be used as a synonym for "decomposable to infinity", terminology that is more acceptance "Perceptual" of the term, as its scientific meaning.
Mathematically, the corollary of the refinement of the scale of observation is the fact that no known Euclidean symmetry is detectable in each fragment studied. Multiple levels of the mesoscopic description, virtually infinite, no longer seem to be correlated hierarchical continues, as the laws of symmetry that generally characterize an object as a whole do not seem to be through the fragmentation of all original. Any fragment appears as a new full, in appearance (that is to say from the point of view adopted) to all foreign landforms which is extracted detail. But even a fractal set contains all the details in a structural relationship defined by its unique dimensional measurement. The law of unity between all morphological fractal chosen by "the observer" and its parts is in no way obsolete, although the perceptual aspect of these details will always be infinitely differentiated and diverse, reflecting Game systematic variations of scale review.
In physical nature, however, compliance levels are not infinite, unlike a mathematical fractal, geometric abstraction with no counterpart in reality. Physicists distinguish among natural objects, the multi-fractal (mostly statistically self-similar objects) of simple fractals (mainly objects with different scales are directly self-similar, or result from each other by affine transformation on model of the famous von Koch snowflake, among countless other possibilities).
Gravity shapes the universe fractal
Interstellar clouds are overlapping entities, similar in structure over several orders of magnitude in size. What is the origin of such fractals? A new theory based on thermodynamic study of self-gravitating medium shows the fruitful analogies with critical phenomena characteristic of phase changes of a physical system. Phenomena that are found on a larger scale, that of clusters and superclusters of galaxies.
Our galaxy, the Milky Way, is composed of a hundred billion stars and hydrogen gas mixed with dust. The interstellar medium is now only a few percent of the total mass of the Galaxy. At the very beginning of its formation, the gas was the major constituent, from which the stars formed by gravitational collapse. Even today, few stars are formed annually in the Milky Way.
The interstellar medium is far from a homogeneous gas: it is distributed in clouds of all sizes, which can be seen in the sky as dark spots in front of the stars, because dust absorbs visible light (see photo above ). The gas clouds are more dense they are smaller and, depending on its density, atomic or molecular gas (H2 or H). In the densest clouds, molecular clouds, the heavy elements formed inside stars (carbon, oxygen, nitrogen, etc..) Combine into molecules, the most abundant being carbon monoxide CO. These molecules emit characteristic emission lines in the millimeter wavelengths. It is they who are the source of our knowledge of the environment, because the main H2 molecule does not radiate and is detectable at very low temperatures prevailing there, the order of 10 Kelvin (263 degrees Celsius).
The molecular lines provide two types of information. They allow, on one hand, to know * Doppler dynamics of clouds and on the other hand, intensity is related to the amount of gas in the line of sight. It was thus possible to connect the mass of clouds with internal velocity dispersion * and with their size. The result is that the mass M of a cloud varies as a power law with size r. In other words, M is proportional to r D, D with non-integer power, roughly equal to 1.7. If the medium was homogeneous, and the clouds all the same density, D is equal to 3, since we are in a three-dimensional space. This non-integer power, less than the dimension of space is characteristic of a fractal structure as Benoit Mandelbrot defined it in 1975.
The structure of molecular clouds is very hierarchical in the sense that the clouds are actually made of smaller condensations, themselves containing small fragments, etc.., Like Russian dolls, on at least five to ten levels. Like any fractal structure is self-similar, that is to say that it is reproduced with the same aspect at all scales. It is thus impossible to guess the absolute size of a cloud observed if we do not know its distance.
The largest observed clouds have a mass of one million solar masses, and their size is about 300 light years. Clouds of larger size can be formed: they are sheared by the tidal forces caused by the Galaxy itself. On the other side of the hierarchy, what is the smallest size observed interstellar clouds? A first limit is given by the spatial resolution of telescopes *, a fraction of a second of arc with the millimeter interferometers *, which corresponds to a size of several hundred astronomical units *. More recently, thanks to the international VLBI (Very Long Baseline Interferometry) telescopes separated by thousands of miles, operating in interferometric mode at a resolution of the order of a millisecond of arc, it was determined sizes ten times smaller Again, the order of tens of astronomical units. Such fragments have a mass approximately equal to that of Jupiter. The hierarchy of clouds is very mixed: the ratio between the smallest and the largest size is about one million, and the mass ratio of one billion.
How such structures could they form? Are they in balance and what is their role in the formation of stars? For a long time, astronomers know that the efficiency of star formation from the interstellar medium is, surprisingly, very low. However, the time of gravitational collapse of clouds is very short, it is 250 years for the smallest fragments up to two million years for giant molecular clouds. If the collapse continued until the formation of stars, it would be impossible to explain the persistence of gas clouds in the Galaxy since the beginning of its formation, that is to say for ten billion years. But at each scale, the collapse is stopped by the agitation of the sub-disordered fragments of the cloud, equivalent to a "turbulent pressure" that-balance against gravitational forces. These turbulent motions are supersonic and highly dissipative: the shock waves they generate, the kinetic energy of the cloud is dissipated very quickly (on the scale of a free-fall time *). Turbulence must be maintained at all times. But by what mechanism? One hypothesis proposed is that the very formation of young stars within the cloud could maintain turbulence in the energy released in various forms (bipolar jets, stellar winds, supernova explosions, etc.).. However, the velocity dispersion observed in molecular clouds form stars is very similar to the quiet clouds, which do not form. Such a solution can not be general.
And so finally, the physical environment was much more simple? The existence of scaling laws in this environment * apparently disordered and chaotic we suggested that a theory based solely on gravity could probably explain the phenomena. A first step was to model the clouds on this principle. The formation of fractal explained by a process of gravitational instability, followed by fragmentation. This process has no characteristic scale and may continue cascading, provided that the gas remains at constant temperature (isothermal), that is to say that he is able to exchange energy by radiation.
A cloud of gas under isothermal conditions, as is the case for the interstellar medium, tends to focus and increase its density in the central parts. But the free-fall time is even shorter than the density is greater, and the cloud becomes unstable when the center becomes too dense to the edge: the cloud fragments into several pieces (typically 5 to 10) more dense, which in turn will focus and so on, recursively. The result is a hierarchy of clouds, becoming denser at every level. This recursive fragmentation stops when the density is so great that the gas becomes opaque to its own radiation. The gas clouds in the center, heated by the beginning of gravitational collapse, can not radiate its heat and drain, and the resulting pressure stabilizes and prevents the collapse and fragmentation. Isothermal regime, the cloud passes an adiabatic regime, that is to say he can not exchange energy with the outside. The smaller fragments provided by this model correspond to the observed structures mentioned above, equal to the mass of Jupiter.
Reached this size, the fragments by collisions merge to form larger structures, and a statistical equilibrium is established between fusion and fragmentation. As a result, the stability of the entire cloud is extended on scales of billions of years. The turbulent motions are continuously maintained and regenerated by gravitational instabilities. The hierarchical fractal structure explains the stability of the clouds.
These early models have thus shown that the assumption that only gravity could be responsible for the fractal structure of the medium was plausible. But curiously no theory for such a set of fragments in equilibrium quasi-isothermal, variable number, subject only to their self-gravitating, had yet been developed. Nous avons donc étudié la thermodynamique du problème, très complexe a priori , puisque toute particule interagit avec toutes les autres. Mais il s'avère que les équations peuvent se simplifier et, surtout, nous avons pu montrer que le système est mathématiquement équivalent à celui d'un ensemble de moments magnétiques (spins), ou à un fluide dont l'état devient critique lors d'un changement de phase. Un des prototypes de ces phénomènes critiques est l'opalescence qui survient à la transition liquide-vapeur d'un fluide au point critique. Des fluctuations macroscopiques de densité se développent à toutes les échelles dans le fluide et réfractent la lumière, ce qui explique l'opalescence. L'étude des phénomènes critiques accompagnant les changements de phase a permis de comprendre toute une série de phénomènes, dès les années 1970-1980. Que ce soit dans des domaines physique ou biologique, les lois d'échelle et la formation de structures self-similaires peuvent s'interpréter de la même façon par des lois universelles, définissant des « classes d'universalité « . En effet, les fluctuations qui se développent au point critique obéissent à des lois statistiques générales, indépendantes des forces microscopiques en jeu, et fonction seulement de la dimension de l'espace et des symétries des forces. Les exposants critiques, reliés à la dimension de la structure fractale obtenue, sont alors universels.
Dans le cas du système autogravitant, la théorie prévoit que le milieu est critique quelles que soient les valeurs des paramètres externes (comme la température). Les fluctuations qui se développent à toutes les échelles, et qui correspondent aux nuages, sont alors prédites par la théorie, de même que la dimension fractale résultante, avec un bon accord avec les observations.
The theory also applies to galaxies taken as a set of self-gravitating points. These form a hierarchical structure in agglomerate into groups, clusters and superclusters. Between the size of a galaxy of about 100 000 light years, and that of larger superstructures observed (in the order of one billion light years) galaxies form a fractal structure of dimension close also D = 1.7. Of course, the fractal is not infinite, as all the actual physical structures, there are lower and upper limits in size. If the lower bound here is the scale of a galaxy, we do not yet know the exact size of the upper bound, but we know that the universe is homogeneous on a large scale, as witnessed by the observation of the bottom of * cosmic background radiation at 3 Kelvin satellite CO BE. This homogenization scale due to the fact that self-gravity of the structures is more prominent with the expansion of the universe. A very large scale, the dimension D will become equal to 3. Already in the existing catalogs of galaxies, we detect a significant increase in D on a large scale, but with considerable uncertainty because it depends on the chosen model of the universe (the distance assigned to each object depending on the curvature of the universe, its density, the Hubble constant, etc.., still poorly known parameters). The scale on which there is a transition to a homogeneous universe has become a hotly debated point, that very large-scale surveys of millions of galaxies could be resolved in the next few years.
