THE THEORY OF CHAOS
Fractals - Chaos theory - Madel brot - The theory of the three bodies - Poincare - The strange attractor - Lorenz Small events raise Chaos does not mean lack of order, but inability 'to predict long-term unpredictability'. In this Quantum mechanics is the basis of almost all of modern physics. I say almost because 'it fails to explain the chaotic events.
Nature shows a tendency towards chaos. The little things we encounter in everyday experience, in weather systems, the drip of a faucet, the plumes of smoke from a cigarette, the heartbeat, the way a flag snapping in the wind or what they are made up in queues Highway ol'andamento of the bag. The charts are linked to the chaos of fractal geometry. The fractal and 'a complex structure that has a particular form, this is repeated components in more 'small, thus showing similarities to internal form. Eg. a fern, or a coastline, a mountain, a cloud, a group of galaxies are natural fractals.
The three-body problem
Fractals
Fractals - Chaos theory - Madel brot - The theory of the three bodies - Poincare - The strange attractor - Lorenz
Jung tells of a girl of 8 years before he died he made many strange drawings, one of them was a drop that was a tree. Put the drop shaft and have a fractal ,
form of a tree that contains a form of tree. This ' a fractal. In our world there is' a drop that contains our world. Imagine a context where there is 'a woman who looks a framework where there is' a woman who looks at a painting ... The microcosm reflects the macrocosm, as they say the alchemists. The fractal shows us that reality 'and' holographic as they have always said the metaphysical or anti Indian alchemy, and all 'in all.
If quantum physics invented 'a language different from that of matter, chaos theory shows us that the world can' be studied with other languages. The old physics was based on the logical mind that has a linear path and tried linear paths in nature, studying the things that were repeated, which could be expected and had regular shapes; stdio the world like a clock. But in this way nature is out, 'cause, in hindsight, nature is not' regular, not 'linear, not presented as fixed and non-geometric shapes and' so 'as predictable as the science demands For this and 'chaos theory was born. It studies systems that have predictable behavior, and the 70 'and' become one of the areas of mathematics more 'interesting, a theory of rare beauty that is highly aesthetic results.
If quantum physics invented 'a language different from that of matter, chaos theory shows us that the world can' be studied with other languages. The old physics was based on the logical mind that has a linear path and tried linear paths in nature, studying the things that were repeated, which could be expected and had regular shapes; stdio the world like a clock. But in this way nature is out, 'cause, in hindsight, nature is not' regular, not 'linear, not presented as fixed and non-geometric shapes and' so 'as predictable as the science demands For this and 'chaos theory was born. It studies systems that have predictable behavior, and the 70 'and' become one of the areas of mathematics more 'interesting, a theory of rare beauty that is highly aesthetic results.
While the old physical demands that nature has a certain order, and is based on Euclidean geometry, which creates smooth textures and measurable, this theory finds instead is a clear trend in the nature of chaos. So if you want to see in depth the nature, the mind should not linear but another mind, another language. We find chaos in many mundane daily events: eg. the dried leaves falls to the ground if (shapes similar to each other), weather systems, the drip of a faucet, the plumes of smoke from a cigarette, the heartbeat, the shape of the clouds, the zigzag of lightning, the branches of trees, the venous or arterial network, the surface of the lungs, the convolutions ... all irregular structures of the brain, where nature can put designs in long narrow spaces.
Eg. wrinkling in the lungs, and can gather in a confined space an area the size of a tennis court. Think of the bronchi branch into bronchioles and then into ... sottobronchi Chaos surrounds us everywhere, in the water vortex, the flow of air that flows through the pipes of an organ, in the foam of the falls, in the gases of Jupiter ...
Chaos and 'the rock we find on the wooded mountain in the moss, which repeats the shape of the wood.
In small and large are repeated in similar forms.
E 'curiously always opposite to the shapes that repeat forms, before the apparent chaos of nature, opposite to the clouds, the waves of the sea, we feel an aesthetic pleasure, while I found no pleasure in watching an upright pole, a highway, the electric wires, all linear human artifacts .. Yet science has not done anything but deny the chaos of nature trying to do something to become linear. But 'cause a wave or a tree or a cloud are beautiful? Why 'are dynamic systems that change over time. The old science was regular geometrical figures, exist in nature, there is not a perfect sphere or an equilateral triangle, are abstractions of the logical mind. In nature there are largely irregular shapes, we think form of a coast. When
and 'tried to mathematize these phenomena, and' discovered a new geometry with shapes that were not measurable in whole numbers rather than with fractions, and these fractions are called fractals.
Fractal mean fractional but also fractus = broken, rough, uneven .. The fractal
and 'a complex, irregular, with a shape that is repeated.
The curious thing is' that no matter how bizarre this form, and 'anything but random, rather it' s so precise that is repeated in detail. I mean ' to analyze any scale I find that form the same structure repeated in all its details,
its parts. Which means that if I take for example. the irregularities 'of the Tyrrhenian coast, and then just take a piece of this coast and enlarge it, I find the same structure, even for most' small again find the same design, and 'as if I had an irregularity' constant, which reappear in the same way. This applies to a fern, a mountain, a cloud, a group of galaxies, a snowflake ... they are all natural fractals. A snowflake (and each and 'unique and different from everyone else) and not just' a unique design and beautiful, but every little part repeats the entire design, in accordance with a holographic.
Chaos and 'the rock we find on the wooded mountain in the moss, which repeats the shape of the wood.
In small and large are repeated in similar forms.
E 'curiously always opposite to the shapes that repeat forms, before the apparent chaos of nature, opposite to the clouds, the waves of the sea, we feel an aesthetic pleasure, while I found no pleasure in watching an upright pole, a highway, the electric wires, all linear human artifacts .. Yet science has not done anything but deny the chaos of nature trying to do something to become linear. But 'cause a wave or a tree or a cloud are beautiful? Why 'are dynamic systems that change over time. The old science was regular geometrical figures, exist in nature, there is not a perfect sphere or an equilateral triangle, are abstractions of the logical mind. In nature there are largely irregular shapes, we think form of a coast. When
and 'tried to mathematize these phenomena, and' discovered a new geometry with shapes that were not measurable in whole numbers rather than with fractions, and these fractions are called fractals.
Fractal mean fractional but also fractus = broken, rough, uneven .. The fractal
and 'a complex, irregular, with a shape that is repeated.
The curious thing is' that no matter how bizarre this form, and 'anything but random, rather it' s so precise that is repeated in detail. I mean ' to analyze any scale I find that form the same structure repeated in all its details,
its parts. Which means that if I take for example. the irregularities 'of the Tyrrhenian coast, and then just take a piece of this coast and enlarge it, I find the same structure, even for most' small again find the same design, and 'as if I had an irregularity' constant, which reappear in the same way. This applies to a fern, a mountain, a cloud, a group of galaxies, a snowflake ... they are all natural fractals. A snowflake (and each and 'unique and different from everyone else) and not just' a unique design and beautiful, but every little part repeats the entire design, in accordance with a holographic.
(Snowflake) 
The fractal geometry reveals beautiful forms that were gradually mathematized with the computer. Benoit Mandelbrot
study 'these complex curves which are organizations of organizations in accordance with rules cosmetics, holographic designs, for which a given form, it is illegal, is repeated in its parts more and more 'small, like a matryoshka. (Fern) 
The science of chaos denies the absolute determinism of nature. The physical
wanted order, regularity and extent ', but the science of chaos instead ennobles its disorder and irregular '. It' s not true that if I knew all the cases, I could predict all the persons, just recently, very little variables and create unpredictable events.
(Cactus) 
Our very life depends on a small irregular choices, if we wake up five minutes later, we leave the house after a pot and 'fell, we take the next bus, and maybe we meet' man of our life. great changes. Chaos theory turns out that, just by changing the initial conditions, we may have totally different outcomes. Here, too, physicists were left shocked.
(Fractal of a coast) 
Yet myths ancient Greeks said that the gods' Primordial were two Order and Chaos.
With the chaos theory have been studied so elusive first new events, such as stock market fluctuations, epidemics, demographic changes, schizophrenia ..
(Fractal tree) 
The human body itself, in the light of this theory, it is different, no more 'as the sum of organs connected mechanically and predictable, but as a connection sensitive to small changes of pace, place of irregular oscillations that are intrinsic constants, however. With the chaos theory have been studied respiratory disorders, cancer cell cycles, severe fluctuations of blood cells, the irregularities' heart, seizures, brain electrical spikes, etc. .. Eg. epilepsy could provoke a slight variation by neuronal electrical impulses, so 'to trigger a compensatory chaos. (bronchogram human) 
Today we are all very sensitive to changes in the stock market, but with the traditional mathematics and 'completely impossible to find equations that allow to make predictions, the slightest change in the stock market may' cause a snowball effect, financial markets are a dynamic system sensitive to disturbances. (Algorithm repeated on a PC) 
Any changes in more 'or less can' produce cascade effects.
With the chaos theory disappears in the sterile determinism that mechanically proximate causes predictable fixed effects, random and accidental elements here Instead, they can actually fit into 'the world changed, the historicized world, becoming enters time . You start to talk about the case.
(Ammonite) 
Chaos theory is paradoxical findings to those phenomena that seem to escape determinism, as the line of a coast or a cloud, the rate of drops falling from a faucet closed mind, I wanted to smoke a cigarette, the time that will '. The same cause-effect relationship here is not 'quantified the old way,' cause a small alteration can 'produce catastrophic changes can not be determined. As they say: "
The beat of a butterfly's wings in the West can 'produce a hurricane in the East . E '
the butterfly effect.
the butterfly effect.
(spiral of Archimedes and sunflower) 
drop a small stone in the mountains can 'unleash a terrifying avalanche. A small change could inadvertently ' have exponential growth. It 's not true that driving with a cue ball pool you can' determine its trajectory, a small difference can 'amplify the point of making absolutely no predictable outcome, and this' just chaos, not determined. meteorology and 'evident in perturbed systems such as the Mediterranean can not be done more weather' long 24 hours. Increasing the time, decreases the possibility 'of certainty. The science of chaos was a precursor in Henry Poincare '
, one of the most' great mathematicians at the end of 1800, but development 'by chance only in 1961 thanks to the American meteorologist named Edward Lorenz
. The famous MIT Institute Technology of Massachusetts, with a huge and primitive computer low memory, Lorenz tried to calculate the movements of the atmosphere, but, having a high number of decimal places, I think 'it was the same round and saw that any rounding, for small that had involved a huge change in the final, so you could not make forecasts beyond 5 or 6 days. Weather fluctuations could still be microscopic undetectable and can amplify up to change the climate of the planet. study time turned the interest of all students on the chaotic phenomena. But when they came back to study the sky, hoping to find the Newtonian accuracy, they discovered that the movements of celestial bodies, which seemed so 'regular, had also been given the irregularities'. For example, the sun does not rise at the same point of the horizon, but goes back and forth. Even the stars are located in different parts of the sky and can disappear for entire seasons. There are even of celestial bodies called 'wanderers', because' going on and then suddenly become 'backward', ie 'come back.
, one of the most' great mathematicians at the end of 1800, but development 'by chance only in 1961 thanks to the American meteorologist named Edward Lorenz
. The famous MIT Institute Technology of Massachusetts, with a huge and primitive computer low memory, Lorenz tried to calculate the movements of the atmosphere, but, having a high number of decimal places, I think 'it was the same round and saw that any rounding, for small that had involved a huge change in the final, so you could not make forecasts beyond 5 or 6 days. Weather fluctuations could still be microscopic undetectable and can amplify up to change the climate of the planet. study time turned the interest of all students on the chaotic phenomena. But when they came back to study the sky, hoping to find the Newtonian accuracy, they discovered that the movements of celestial bodies, which seemed so 'regular, had also been given the irregularities'. For example, the sun does not rise at the same point of the horizon, but goes back and forth. Even the stars are located in different parts of the sky and can disappear for entire seasons. There are even of celestial bodies called 'wanderers', because' going on and then suddenly become 'backward', ie 'come back.
(The apparent motion of a planet) 
Man has always sought security in the knowledge of the positions of the sun, moon and stars main. See Stohnengen stones, temple and 'a great cosmic clock that marks both the solstice the sun is the lunar month, the movement of the Moon and' even more 'twisted than that of the sun, and to date, with all our observatories, telescopes and computers, we have not yet been completely determined. (clumps of salt to the surface of a pot of water left in the fridge) 
Nature seems to behave arbitrary and not at all mechanical, so you can speak 'of' creative nature '. Nature shows a tendency towards chaos. The little things we encounter in everyday experience, in weather systems, the drip of a faucet, the plumes of smoke from a cigarette, the heartbeat, the way a flag snapping in the wind or what they are made up in queues Highway ol'andamento of the bag. The charts are linked to the chaos of fractal geometry. The fractal and 'a complex structure that has a particular form, this is repeated components in more 'small, thus showing similarities to internal form. Eg. a fern, or a coastline, a mountain, a cloud, a group of galaxies are natural fractals.
(Galaxy) 
The fractal geometry reveals beautiful forms and has found application in computers. Benoit Mandelbrot studio'queste complex curves show that organizations of organizations in accordance with aesthetic rules of nature, a kind of holographic seal poured on the inside so that a given form repeats the same form as in its parts more and more 'small. The science of chaos denied the absolute determinism of nature. The order had to be physical, physics accept only what 'that was lawful, but the science of chaos introduced disorder and irregular '. (Omino Mandelbrot) 
Madelbrot was a Polish mathematician whose family was all made up of mathematicians, being sheltered jew 'to Paris to escape the Nazis. Discover 'fractals by accident while doing research for IBM in 1979 and applies' to the economy so as to undermine the foundations of classical economics and other of modern finance, giving rise to a fractal and finance and he also un'ecofisica understand that neurology would know a total rebirth. Eg. in neurology and is' seen the old shows less branching of Purkinje cells, and a reduction in their fractal dimension.
Chaos theory gives rise to holistic images, and this' is the opposite of determinism. There are no cases related to events, but the forms that arise out of all the parties. Initially
calculations to study these forms were too complex, and only after the '70s, when computers were used chaotic systems reveal their beauty. Science made leaps forward through instruments such as the microscope to study biology or the telescope to study the stars: the computer and the accelerator of particles. The chaotic events became the subject
science since 1980 and there were physicists like Hawking
who sought a system that encompassed both the theory of relativity 'that quantum mechanics.
events show the chaotic nature that second dance choreography constant.
calculations to study these forms were too complex, and only after the '70s, when computers were used chaotic systems reveal their beauty. Science made leaps forward through instruments such as the microscope to study biology or the telescope to study the stars: the computer and the accelerator of particles. The chaotic events became the subject
science since 1980 and there were physicists like Hawking
who sought a system that encompassed both the theory of relativity 'that quantum mechanics.
events show the chaotic nature that second dance choreography constant.
The three-body problem The problems became more 'complex when it was discovered' that the relationship between body alters the context if they were more 'than two. Newton had started from the observation of celestial bodies and tried to establish their movements in accordance with their respective mass and gravitational force ', so the problem of determining the orbits of three celestial bodies does not seem very difficult in mathematical astronomy. But what with the two bodies and 'easy, with three gets more complicated.
Take the earth and the moon that revolves around attracted by its gravitational force 'but together with his resists. The arrival of a third body, the astronomers say that it is' a 'disturbance'. With three bodies
mathematical calculations can not reproduce what 'really happens and are always a gap, in fact we have no means a simple perturbation, we have a new problem.
An easy example would be that of a married couple living together for a long time and have a mutual balance, sometimes comes to their view that the twenty nine year old daughter 5 years living away from home, what happens is not 'a simple disturbance but a totally new relationship in which even the relationship between the couple goes topsy-turvy. The intrusion of a third person in the couple causes a lot more 'damage that a simple noise. If you imagine a husband and wife should love and in agreement between them can put a mother-in-law have the same disaster.
not for nothing that Jesus' said, "Where are
more 'than two, I will be'
with you." We see that there was no need!
So despite all the calculations possible, the triad Sun-Earth-Moon could not be explained. Even EULERO
step ' entire life to studying the three-body problem and finally had to give up.
needed something to get things moving again, Newton's equations were far from sufficient. Newton was convinced that enough to know the position and velocity 'of an object to predict its trajectory all its future data, but it was not so'. Mathematicians had in front of a set of values \u200b\u200band believed that the series were infinite in the same way. Given the first data series, thought it was easy to predict the next, but it was not so '. The series was called 'converged' cause was to converge to a final result but no. It was his genius that was Poincare '
the end of 1800 with the three-body problem in understanding the evolution of certain physical systems and 'so' sensitive to initial conditions that it becomes impossible to predict its future behavior.
mathematician was a competition in honor of the King of Sweden and launch 'a challenge to mathematicians around the world to determine the three-body problem. In short, it was evident that examining only one orbit and another was not enough, the problem does not solve some of the parties. But
Poincare 'had the brilliant idea of \u200b\u200breversing the problem. Instead of isolating fragments of reality 'with differential equations, I try to' grasp the whole of its globality ', reversing the traditional process of the old physics. need to find a new paradigm.
The first step was to focus on the whole instead of the parties .
Poincare 'produced an invention that is the basis of the geometric theory of chaos. We are in a three-dimensional space, a ball can 'go left or right, up or down. But Poincare 'think' that could assume a multidimensional space in which there were at the same time, three-dimensional coordinates of the body and three-speed '(top to bottom, from right to left, etc..) So to describe the ball use on the move ' 6 dimensions, so 'sizes 6 to describe the Moon, the Sun 6, 6 for the Earth. Each system occupies the position of a point, but the system moves, and 'dynamic, that' the trio Moon-Sun-Earth moves in the sky. Poincare 'look' at how the whole system rotating shifts and find out 'which made him the second a beautiful figure. Instead of studying the movement of each body, the study 'the global movement of the trio and find out' who the stages traveled drawings fixed, so 'invented' the term
phase space.
The total movement form of the spirit that created
the shape of a butterfly's wings . This will study 'the system in its future evolution over time. When the system undergoes a change, the point in the multidimensional space that represents a curve moves. could be the movement of a leaf that is detached from the branch and maneuvered in the air. If the leaves are two, and that each could be 'very sensitive to minor variations, so it falls to the ground and the other in a river.
Take the earth and the moon that revolves around attracted by its gravitational force 'but together with his resists. The arrival of a third body, the astronomers say that it is' a 'disturbance'. With three bodies
mathematical calculations can not reproduce what 'really happens and are always a gap, in fact we have no means a simple perturbation, we have a new problem.
An easy example would be that of a married couple living together for a long time and have a mutual balance, sometimes comes to their view that the twenty nine year old daughter 5 years living away from home, what happens is not 'a simple disturbance but a totally new relationship in which even the relationship between the couple goes topsy-turvy. The intrusion of a third person in the couple causes a lot more 'damage that a simple noise. If you imagine a husband and wife should love and in agreement between them can put a mother-in-law have the same disaster.
not for nothing that Jesus' said, "Where are
more 'than two, I will be'
with you." We see that there was no need!
So despite all the calculations possible, the triad Sun-Earth-Moon could not be explained. Even EULERO
step ' entire life to studying the three-body problem and finally had to give up.
needed something to get things moving again, Newton's equations were far from sufficient. Newton was convinced that enough to know the position and velocity 'of an object to predict its trajectory all its future data, but it was not so'. Mathematicians had in front of a set of values \u200b\u200band believed that the series were infinite in the same way. Given the first data series, thought it was easy to predict the next, but it was not so '. The series was called 'converged' cause was to converge to a final result but no. It was his genius that was Poincare '
the end of 1800 with the three-body problem in understanding the evolution of certain physical systems and 'so' sensitive to initial conditions that it becomes impossible to predict its future behavior.
mathematician was a competition in honor of the King of Sweden and launch 'a challenge to mathematicians around the world to determine the three-body problem. In short, it was evident that examining only one orbit and another was not enough, the problem does not solve some of the parties. But
Poincare 'had the brilliant idea of \u200b\u200breversing the problem. Instead of isolating fragments of reality 'with differential equations, I try to' grasp the whole of its globality ', reversing the traditional process of the old physics. need to find a new paradigm.
The first step was to focus on the whole instead of the parties .
Poincare 'produced an invention that is the basis of the geometric theory of chaos. We are in a three-dimensional space, a ball can 'go left or right, up or down. But Poincare 'think' that could assume a multidimensional space in which there were at the same time, three-dimensional coordinates of the body and three-speed '(top to bottom, from right to left, etc..) So to describe the ball use on the move ' 6 dimensions, so 'sizes 6 to describe the Moon, the Sun 6, 6 for the Earth. Each system occupies the position of a point, but the system moves, and 'dynamic, that' the trio Moon-Sun-Earth moves in the sky. Poincare 'look' at how the whole system rotating shifts and find out 'which made him the second a beautiful figure. Instead of studying the movement of each body, the study 'the global movement of the trio and find out' who the stages traveled drawings fixed, so 'invented' the term
phase space.
The total movement form of the spirit that created
the shape of a butterfly's wings . This will study 'the system in its future evolution over time. When the system undergoes a change, the point in the multidimensional space that represents a curve moves. could be the movement of a leaf that is detached from the branch and maneuvered in the air. If the leaves are two, and that each could be 'very sensitive to minor variations, so it falls to the ground and the other in a river.
(crystals of manganese oxide) 
was at that point that Poincare 'deals' of aid from the king of Sweden. Every heavenly body moves in a stream of mathematics.
P. I guess' a vertical wall that dissect every move. If a body back and forth in a periodic manner, the section had a point, but if a body moves in an ellipse on the section had a coil. If a celestial body had repeated its orbit, the section would come the same spirit. But if its orbit is moved, the section would show different figures. Poincare 'discover' that the orbits were moving but gave rise to similar figures. This' chaos. (Neutrofine, small protein secreted by the nervous system of the brain) 
He saw clearly that a minimum of change in position or velocity 'of the three bodies totally changed the orbit, the system ceased to be stable and plunged into chaos (as the couple arrived in the house when her daughter or mother-in-law and then can 'happen all). A system perfectly steady and predictable as the three bodies could lead to unpredictable and undefined, Poincare 'had discovered the uncertainty
.
In a system of three bodies orbit a planet follows a complex and unpredictable, that is' chaotic. Newton had noticed 'cause had prevailed in him the desire for harmony and order. Suddenly, the sky had turned into a perfect watch well-ordered and rigid. The message that I received three centuries of science and 'the world' order and woe betide anyone who contradicts it. Yet Newton was not able to dominate the moon, rebellious to all calculations, the moon dance. Poincare 'had discovered the dance of the moon, not around the Earth in the universe, its choreography.
The metaphor of modern physics became the
DANCE.
Bateson said: "We are used to imagine how things fixed structures (except those music). But this 'and' nonsense. A structure is 'a dance of interacting parts, where the structure determines the process, but the process can' determine the structure .
In a system of three bodies orbit a planet follows a complex and unpredictable, that is' chaotic. Newton had noticed 'cause had prevailed in him the desire for harmony and order. Suddenly, the sky had turned into a perfect watch well-ordered and rigid. The message that I received three centuries of science and 'the world' order and woe betide anyone who contradicts it. Yet Newton was not able to dominate the moon, rebellious to all calculations, the moon dance. Poincare 'had discovered the dance of the moon, not around the Earth in the universe, its choreography.
The metaphor of modern physics became the
DANCE.
Bateson said: "We are used to imagine how things fixed structures (except those music). But this 'and' nonsense. A structure is 'a dance of interacting parts, where the structure determines the process, but the process can' determine the structure .
Fractals The same shape was found by Lorenz, a meteorologist who was studying the movement of hot and cold. Here too the point never to repeat the same movement, but similar movements, as if there was a constant form that attracted each loop, and coined 'the term strange attractor . In short, came a series of similar designs, very beautiful, in which points are not scattered at random but were 'lured' by precise curves, such as iron filings and 'attracted by a magnet forming two curved like the horns of a bull or the atmosphere as the particles are distributed over the poles as ram's horns.
(strange attractor of Lorenz) 
The chaotic motion of the point was because 'no one could predict where he would find after a while, but eventually showed similar forms. At this point we noticed another oddity: not only created a certain form similar to the previous one, but if you watched with a microscope, a detail it was discovered again in the same small form, and again and again, according to ; 
autosomiglianti drawings.
Eg. the strange attractor of the orbits of stars in the shape of a banana. If you enlarge a tract is split again into a banana, and so 'on. autosomiglianti drawings. (strange attractor of Rossler) 
Even in meteorology Lorenz I ' of strange attractors. The complex movements of air masses that move causing fine weather or rain can be represented with a point and the dance form that comes up and 'the two wings of a butterfly. But every point is divided into two, and then again in two and so on. like a matryoshka, infinity, the finite space contains infinite forms, each like the previous ones, each side and 'formed as two wings of a butterfly. was impossible to measure with a whole number of these irregular shapes, were used fractional numbers, and they were called fractals.
The strange attractor has a fractal shape.
The 'fractal' designates the number of objects whose size can 'be expressed only as a fraction. Euclidean geometry fails on irregular objects, but they are the majority of things we see. Nature does not give us 'but precise geometric shapes and' irregular. Yet in its irregularity 'has the cadences of the phases. The language of logic and rules of science leave out all over the world, needed a new language.
Professor. De Rosa (professor of business administration) said: "
Escape from logic and 'extravagant delusion. Be subservient to the logic and 'streamlining delirium. We need rationality 'softened in contact with the empirical, where logic does not trump it' yields, which leads us to the limits of the intellect and on the borders of reality.
And that dialogue with the poetry
.
The 'fractal' designates the number of objects whose size can 'be expressed only as a fraction. Euclidean geometry fails on irregular objects, but they are the majority of things we see. Nature does not give us 'but precise geometric shapes and' irregular. Yet in its irregularity 'has the cadences of the phases. The language of logic and rules of science leave out all over the world, needed a new language.
Professor. De Rosa (professor of business administration) said: "
Escape from logic and 'extravagant delusion. Be subservient to the logic and 'streamlining delirium. We need rationality 'softened in contact with the empirical, where logic does not trump it' yields, which leads us to the limits of the intellect and on the borders of reality.
And that dialogue with the poetry
.
(Example of Julia sets) 
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