"Fractal geometry is not just a chapter of mathematics, but one that helps everyone see the same world differently."
(Benoît Mandelbrot)
"Fractals are the simplest means of creating complexity" (Jorge Wagensberg).
Fractal Geometry
Definition of fractal
There is no precise and formal, unanimously accepted definition of the concept of fractal. The most common definition is intuitive: a fractal is a hierarchical geometric object that includes itself a certain number of times at each level, or a geometric object that contains a pattern that repeats itself indefinitely.
Cantor set (generation)
Koch's curve open (generation)
Koch's curve closed (generation)
Fractal objects are geometric shapes resulting from the application of certain simple generating laws that are applied recursively or iteratively at each level, so that the result, which may be complex, reflects spatial homogeneity, scaling, and self-similarity. A self-similar set is one that can be decomposed into parts, each of which represents the total set. Similar figures are those that have identical shape, although they may differ in size, spatial location and orientation.
Famous self-similar fractal sets are: the Cantor set (also called "Cantor dust") and the Koch curve in its open and closed versions (see figures). To generate the Cantor set, a segment is taken and divided into three parts. The central segment is eliminated. With the remaining two segments the same procedure is applied, and so on to infinity. The generation of the Koch curve is based on replacing one segment by four equal segments.
replacing a segment by four equal segments of length 1/3, as shown in the figure, and to each of them the same procedure is applied again, and so on to infinity.
Other examples of fractals are the Peano and Hilbert curves. Both are continuous curves that pass through all the points of a square. The Hilbert curve is a variation of the Peano curve.
Peano curve (generation)
Hilbert curve (generation)
Fractal structures and phenomena
Fractals are found everywhere, in nature and in all kinds of dynamic processes: in the morphology of trees, in flowers (lotus), in plants (cauliflower, fern), in the formation of rocks, in the human circulatory system (veins, arteries, capillaries), in the nervous system, in the folds of the brain, in the fluctuations of the stock market, in the structure of galaxies, in business, in art, in biological rhythms, in lightning, in porous formations, in coasts, in borders, in Brownian motion, in the evolution of populations, etc.
There is a fractal order, more or less hidden, underlying natural phenomena. In fractals there is recursive self-similarity (or self-similarity), that is, the same pattern repeats itself at all levels. The fractal contains itself, refers to itself: in each part is the whole, and the whole is in each part.
Cauliflower Romanescu
Fern
Regular and irregular fractals
Fractals can be regular (such as Cantor's and Koch's) or irregular, depending on whether or not the same structure is preserved, respectively, at different levels.
A random fractal is an irregular fractal generated by some random parameter. A random fractal is, for example, the Cantor set, but eliminating in this case a central segment of random length, repeating the operation on the other two segments, and so on.
Fractals in nature are more or less irregular. Brownian motion is very irregular. A mountain is a random fractal. Ferns, in particular, have almost perfect fractality.
Characteristics of fractals
Fractals are transcendental type objects, since they unite opposite or dual characteristics, integrating the two modes of consciousness (those associated with the cerebral hemispheres):
They unite part and whole.
There is self-similarity, that is, the same graphic pattern (the whole object) is repeated at different levels or scales. There is reflection, since a fractal refers to itself.
Self-similarity is a universally extended property in a great variety of phenomena: climatological variations, formation of coral masses, flows in turbulent regime. Self-similarity can be deterministic or random. The self-similar fractals in nature are usually random.
They unite the finite and the infinite.
For a fractal to contain itself is only possible if there are infinite levels. At the arithmetic level it is also true that infinity is fractal because it contains itself: (∞ =: ∞+1). For example, the Koch curve starts from a finite segment and becomes a curve of infinite length, because at each recursion the length increases in the ratio 4/3.
They unite simplicity and complexity.
"Fractals are the simplest means of creating complexity" (Jorge Wagensberg). There is simplicity in the deep, although externally, superficially, they show great complexity. Indeed, fractals are generated by simple rules. For example, from the simple recursive expression z = z2 + c, where z is a complex variable and c a complex constant, by successive iterations, one arrives at something extremely complex, in this case the so-called "Mandelbrot set", a set that marks the boundary between order and chaos. However, the complexity is organized or ordered, because the same structure of connections unites the different scales.
They unite theory and practice.
Fractals are both a mathematical theory and a method for modeling a wide variety of phenomena.
They unite space and time.
Fractals appear as static structures and as dynamic structures or processes (such as plant growth). Fractal objects are objects that unfold in space-time in a harmonic way, so that the process of evolution is fractal because it relies on a fractal structure, optimizing the number of connections in the minimum space. "Fractality primarily colonizes. It is a way of filling space, it is a way of growing" (Jorge Wagensberg).
They unite order and chaos.
The fractal forms of nature are very varied and, in general, they are not regular. Nor do they respond to complete chaos. Natural forms follow what has been called "organized chaos", a geometrically ordered chaos.
They unite freedom and determinism.
Freedom resides in the fact that the generating laws can be freely chosen. Once chosen, applied recursively, the complete drawing is generated deterministically.
Unites the real and the imaginary.
The most complex fractals (such as the Mandelbrot set) occur in the complex plane. It is there that the imaginary becomes real, tangible, experiential.
They unite or connect geometric spaces of contiguous dimensions.
One of the characteristics of fractals is their fractal dimension D, which measures their degree of occupancy of space. In general, it is a non-integer number. For example, the fractal dimension of the Koch curve is an irrational number, of value approximately 1.1618, i.e., it is between 1 (the dimension of the line) and 2 (the dimension of the plane). And the fractal dimension of the Cantor set is another irrational number, approximately 0.6309, connecting in this case the point (dimension 0) and the line (dimension 1).
They unite mind and nature, inner world and outer world, the particular and the general, conscious and unconscious, science and humanism (including spirituality).
Contemplating a fractal generates a double movement of consciousness: seeing the particular in the general and seeing the general in the particular. They connect us with the depths of ourselves and of all reality, with universal consciousness, pure consciousness, the source of everything, the field of all possibilities, of all meanings, where everything is connected, where the essential unity of all things resides.
In addition to being a science, fractals humanize. When perceived (at a superficial level) they stop time and stop the mind, interiorizing us towards our true being, where there is order and beauty and where there is neither space nor time. "Contemplating a fractal form impresses. One experiences mental joy" (Jorge Wagensberg). Fractals harmonize and even heal. By connecting us with our inner self, thoughts calm down, producing order and balance.
Fractals have brought about a new way of contemplating the world, making us more aware of our external and internal nature. Fractals are consciousness because they are recursive, for recursion is one of the characteristics of consciousness.
Precisely because fractals have a non-integer dimension, we can consider that they connect the conscious and the unconscious. Indeed, rational numbers (which include integers) belong to the external, superficial, conscious world. Irrational numbers belong to the inner, deep, unconscious world.
Moreover, in consciousness there is a fractal resonance. All states of consciousness are resonant states. Resonance occurs when different simultaneous levels interact and reinforce each other. When we perceive an object (superficial), its different aspects (color, size, shape, etc.) connect us with general concepts, and these in turn with other even more general concepts, and so on, in an ascending process towards the universal and profound. In turn, the general concepts of each level connect or resonate in our consciousness with other particular concepts. These connections are manifestations of consciousness. The deeper and wider this network of connections, the greater the consciousness.
As a conclusion, we can say that fractals integrate the two modes of consciousness, by evoking the hierarchical structure of reality from the deep to the superficial level, resulting in or producing greater consciousness, for consciousness is the perception of opposites. With fractals the world is seen in a different way, in a deeper way, where everything is connected.
The Fractal Paradigm
The phenomenon of fractals is a very important discovery-invention, to the point that it is considered something that transcends geometry to become almost a panacea, a universal paradigm, a new and unifying way of seeing reality, a new consciousness that connects us with the essence of all things, since it is stated that everything is fractal: nature, the universe and the mind.
Indeed, many external forms in nature seem to follow fractal patterns or features. Whenever it is necessary to maximize a surface area while minimizing mass, nature uses fractal structure, as in the branching forms of nerves, lungs, and trees. "Fractals colonize" (Jorge Wagensberg).
However, the fractal is only one aspect of nature. As Mandelbrot says, "the geometry of nature has a fractal face". The fractal is an aspect or dimension of nature. There are, evidently, phenomena that are not fractal.
However, if we generalize the concept of fractal and consider or call fractal everything that can be generated by a simple set of laws or simple patterns that, when applied recursively, produce complexity, then we can affirm that everything is fractal, because behind the diversity of phenomena are hidden the same simple principles and the same laws.
The fractal mind
The inner nature (i.e., the human mind) also seems to be configured around certain relatively simple patterns, by some universal laws, resources or semantic patterns that apply always and at all levels. This is why it can be said that the mind is fractal in nature and that the fractal is a metaphor for the mind.
There is currently a clear tendency towards a fractal conception of the mind, justified for two reasons:
By connection or analogy with the role of fractal structures in nature. There is correspondence between mind and nature, between internal world and external world: between the laws that govern the universe and the laws of the mind. The human being has a fractal mind. That is why we tend to create fractal structures spontaneously.
Because the mind is also part of nature and is a reflection of it. According to Pythagoras, the structure of the mind is the structure of the world.
Indeed, the mind shares characteristics of fractals:
It is a complex dynamic structure and, at the same time, organized (or self-organized). There is harmony between these two seemingly contradictory concepts. There is a natural order that is "organized complexity".
Complexity arises from the recursive application of simple generative laws. The same universal laws apply at every level. Complexity is thus apparent, for there is underlying simplicity.
It is a highly interconnected structure at all levels that facilitates the free flow of information. This great wealth of interconnections at different levels allows it to have a network in which there are different alternatives in connectivity, that is, a wide margin of freedom.
It is a dynamic adaptable structure, where new connections are created, existing ones are reinforced or weakened, etc. In short, it allows us to evolve as a living organism. Consciousness is directly related to the quantity and quality of these relationships or connections.
It is a structure that allows to achieve an optimization of the available resources and with the maximum compression (minimum space). The analogous physical image would be that of the brain (maximum compression) and that of the nervous system (minimum number of contact points).
These characteristics are fundamental for mental health, as they allow us to adequately approach life's multiple situations with a wide variety of resources and options.
The fractal collective unconscious
According to Jung's theory of the collective unconscious, there is a collective mind formed by a deposit of archetypal elements, the result of a long process of evolution of humanity. These archetypes, rather than detailed and concrete structures, are patterns, schemes, qualitative relationships. "Archetypes are forms without content" (Jung).
This great repository of our ancestral memories is structured as a network, with multiple connections, with organized and optimally configured complexity. The most reasonable hypothesis is that this structure is an immense shared fractal.
With the process of evolution of mankind, the collective unconscious mind becomes more and more complex, more interconnected (with more relationships). An analogy that illustrates this theme (albeit on a real, conscious level) is the Internet, a highly interconnected repository of information.
The fractal universe
The distribution of matter in the universe follows a pattern as a result of the Big Bang (the beginning of time) and scaled during 14 billion years of cosmic expansion. Cosmology is based on the assumption that, when we look at the universe at the largest scales (over 300 million light-years), matter is uniformly distributed in space. Cosmologists call this a homogeneous (smooth) structure. But these ideas are being challenged and may shake the foundations of cosmology, as a group of researchers [Gefter, 2007] claim that the structure of the universe is fractal, both on large and small scales, with the same patterns repeating to infinity. The same pattern is repeated in solar systems, galaxies, galaxy clusters and superclusters. However, this theory questions Einstein's theory of general relativity and the hypothesis of the uniform growth of the universe from the Big Bang.
In general, the fractality of the universe could explain the deep and hidden structure of the universe and its different levels of manifestation. And, therefore, its ultimate meaning, its essence, which is fundamentally mathematical:
The distribution of matter in the universe, including dark matter. Dark matter is an invisible form of matter that surrounds galaxies and accounts for 80% of the total mass of the universe.
Dark energy. It is admitted in cosmology that the expansion of the universe is due to the so-called "dark energy", so called because it is an energy that we do not see. The explanation may be that this energy resides at a deeper level of physical reality, at an unmanifest level, but perhaps it is the support of manifest matter-energy itself.
The collapse of space-time that occurs inside black holes.
Gravity. According to Dan Winter, "fractality creates gravity."
The imaginary mass associated with tachyons, since fractals are closely related to complex numbers. A tachyon is a hypothetical particle capable of moving faster than the speed of light.
The spiral shape of the arms of spiral galaxies, since the spiral is a type of fractal.
The consciousness of the universe, as fractals represent or symbolize consciousness.
Fractal time
Time is fractal because it includes itself, that is, in every instant there is the totality of time and every scale of time reflects the totality of time: the present, the past and the future. It is like the infinite number and the continuum, which include themselves.
Time is an aspect of consciousness and has two poles (and, at the physical level, associated with the cerebral hemispheres):
External time.
Corresponds to our perception of the external world. We perceive it in a linear way, it is superficial, fixed, finite and objective (common to all human beings). It is associated with the past, memory, the closed and the real.
Internal time.
Corresponds to our perception of the inner world. We perceive it in a non-linear, circular way, it is deep, flexible, infinite and subjective. It is associated with the future, the open, the possible and the imagination.
At the superficial (conscious) level, time manifests itself in a linear form. At a deep level, time does not exist. Einstein said: "Time is an illusion", an illusion of our superficial mind. But time is flexible. It can expand if we go deep. At the limit, in pure consciousness, time disappears. That is why it is said that "Time is not had, it is created". It is created from the deep to manifest at the superficial level.
In Western culture time is considered linear. In Eastern culture it is accepted without question that time is circular (or cyclical). Circular time symbolizes the indivisible unity of time, eternity, where there is no beginning and no end.
The concept of circular time is very old. Its renewed formulation is due to Nietzsche, with the "myth of the eternal return", an idea raised in "The Gay Science" and developed in "Thus Spoke Zarathustra". For Nietzsche, what repeats itself are not only events, but also thoughts and emotions.
The present does not exist because it has no temporal extension. The present is an abstraction, it does not really exist, as the geometric point does not exist. What we call present is a sensation produced by the persistence of events in our memory.
External time is a physical magnitude. Since the appearance of Einstein's theory of special relativity, we know that space and time go together, that it is necessary to speak of space-time. Attempts have also been made to solve certain problems of theoretical physics using the concept of circular time. The most prominent is that of Gödel, who gave a new interpretation to Einstein's theory of relativity.
Time and mind are connected, they are inseparable. Time is really an illusion, a construct of the mind, a concept created to interpret reality. Past and future have no reality of their own. The only thing that remains is the present continuum. The mind perceives time because it is synchronized with the exterior, with phenomena, but on an internal level time is diluted, it disappears.
But it is possible to "free" oneself from time. The key consists in living permanently in the present, because in the present time does not exist. This is what Eckhart Tolle calls "The Power of Now". Through awareness of the now, liberation from time is achieved. "Enlightenment is a state of wholeness in which you are 'unified,' and therefore you are at peace" (Eckart Tolle).
Indeed, when the mind is connected to the past (memory) or to the future (imagination), the mind is in activity. When the mind is connected with the present, time stops and the mind (being synchronized with time) also stops, it stops. It is what Castaneda calls "stopping the world". It is then that one has access to the inner Self, the deep self, a timeless state of consciousness, pure awareness, the source of thoughts, the absolute, the undifferentiated, the unified field of consciousness, the unmanifested, the place where everything is connected, the source of creativity, freedom and happiness, and where one truly experiences the flow of life. "The Self is the One, eternal, ever-present Life" (Eckhart Tolle).
The inner Self cannot be grasped with the mind because it is beyond the mind and thoughts, so it can only be accessed when the mind is stilled. The Self is hidden behind the active mind. If we stop the mind, then we have access to the Self. "To be identified with the mind is to be trapped in time" (Eckhart Tolle).
The general strategy or technique, to access the inner Self and stop time and mind, is to direct attention, at all times, to perceptions, both internal and external. In effect, the mind basically emits (thoughts) or perceives (sensations). The awareness of sensations stops the mind. Conscious perception is the tool to bring the mind to the present. The key is to perceive, to observe without analyzing (past) and without imagining (future), focusing only on the process of perception, on contemplation. By opening ourselves to perception, the mind, body and emotions relax.
Hence the role that the perception of fractals plays in the stopping of time. And, therefore, in the internalization and consciousness.
The fractal, archetype of consciousness
The fractal is a universal archetype, a structure that transcends particular forms, that is beyond concrete contents. And that archetype is self-similarity, a type of reflection that constitutes the foundation of consciousness. In this circular or reflective structure or form, time is annulled and one enters into no-time, into eternity, where the Self dwells. It is a dimension that is beyond particular concepts to become pure consciousness, a consciousness that is beyond the analytical mind and that allows us to perceive the unity of all things and to access a deep, transcendent, true, essential level of wisdom, power, freedom and creativity.
The fractal is the unifying structure of reality. The fractal reaches directly to the consciousness, it is "food" of the consciousness. The message of fractals is consciousness.
Applications of the fractal paradigm
The concept of fractal has been and is one of the most fertile concepts in science, which is having a practically universal application, in all fields It allows modeling and exploring many phenomena and even human activities. Here are some of the applications:
Fractal antennas. A fractal antenna is an antenna designed in a self-similar way to maximize its length with respect to a given surface. In addition to being more compact than conventional ones, they are capable of operating at different frequencies simultaneously. In 1999, it was discovered that self-similarity was one of the necessary requirements for antennas to be invariant (have the same properties) over a range of frequencies.
Fractal geometry is the natural language of chaos theory and nonlinear systems, such as atmospheric weather, fluid turbulence, liquid-solid and gas-liquid phase transitions, etc.
Fractal music. Some examples: Sergent Pepper (The Beatles), traditional Japanese music, Indian regas, folk songs of the Russian travels, jazz, medieval music, Bach, Beethoven, Erik Satie, etc.
Fractal painting. For example, in Escher, Dali, Jackson Pollock and Max Ernst.
Fractal enterprise. It is the application of the fractal paradigm to the world of business and, particularly, to the world of manufacturing. The original idea, developed in 1992 by Hans-Jurgen Warnecke, is based on the principles of self-similarity, self-organization, dynamicity, decentralization and cooperation.
Fractal image compression and landscape generation using fractal algorithms. They are very efficient, saving memory space and processing time.
Fractal architecture is a new paradigm of architecture based on the application of the principles of fractal geometry. Fractal architecture, despite being a reflection of the current culture of new information technologies (widely spread), has a certain mysterious and secret air, as it happened in the past with the Gothic cathedrals and the Masons. But in a certain way it links with sacred architecture, since it connects with forms that affect consciousness. Sacred geometry is a metaphor for the ordering of the universe, that is, the study of the proportions, patterns, codes and symbols that underlie all manifestations of creation. It is the genesis of all forms and a way to understand our true nature.
MENTAL, a Fractal Language
The fractal paradigm may seem at first to be just another paradigm, a way of structuring information like the functional paradigm, the objectual paradigm, etc. However, the fractal paradigm, because of its close relationship with the subject of consciousness and its many universalist characteristics, makes it something special and profound, in such a way that we can affirm that it is a universal paradigm if we consider it as a paradigm based on simple initial concepts applied recursively. In this sense, there are parallels with MENTAL. Here are the points of coincidence:
Generative simplicity.
Fractals always use the same simple generative laws at any scale. MENTAL always uses the same primitives, also simple, at any scale.
Complexity of the result.
The result of applying simple laws recursively or iteratively produces complex structures. The complexity is only apparently, superficial, because the deep refers to simplicity.
Reflection.
Fractals are reflexive, they refer to themselves. MENTAL is a reflexive language. Reflection is one of the characteristics associated with consciousness.
They connect mind and nature, inner and outer worlds. They increase consciousness, and awaken our intuition.
There is maximum freedom and creativity. These two concepts go together. In fractals there is freedom in the choice of the generating laws, with the consequent creativity. In MENTAL, the primitives are precisely the degrees of freedom of consciousness. And creativity comes from the deep level connectivity of semantic primitives.
It allows to express and connect the finite and the infinite. Fractals connect the superficial and finite with the deep and infinite. MENTAL allows to express the finite and the infinite, the latter by means of descriptive expressions.
Besides being a fractal language, MENTAL allows to specify fractal expressions (and, therefore, of infinite type) in a very simple and direct way, by recursion and using potential substitution.
As a conclusion, we can state that MENTAL is a fractal language, in the generalized sense (the same linguistic resources are applied at all levels), integrating the two poles of consciousness, like fractals. But MENTAL goes beyond the fractal paradigm, as it has more semantic resources and possibilities such as: non-local links, sharing, virtual expressions, etc.
Addenda
A brief history of fractal geometry
Fractals have been represented in art and architecture, more or less consciously, in all cultures throughout history. The concept of fractal can be traced back even to Aristotle, but ideas in this field are considered to have started with Cantor and his famous set (Cantor's dust), described in 1883, followed by Peano's curves (1891) and Hilbert's curves (1892).
At the beginning of the 20th century, Henri Poincare, Pierre Fatou and Gaston Julia studied complex dynamical systems, where irregular shapes appear. Fatou and Julia studied the recursive complex function z = z2 + c, finding that for any value of c, the associated set is only two types: connected (consisting of a single piece) or completely disjoint (consisting of a cloud of scattered points). Julia found that it suffices to study whether the point z=0 diverges or not to deduce whether the whole set is connected or disjoint.
Gaston Julia wrote in 1918 (at the age of 25) "Mémoire sur l'itération des fonctions rationnelles" (Memoir on the iteration of rational functions), a voluminous 199-page article, a masterpiece semi-forgotten until it was recovered decades later by Mandelbrot as the theoretical basis of fractal geometry. In that article, he discussed the iteration of a rational function, a subject that was also studied, simultaneously and in a similar way, but from a different perspective, by another French contemporary of his, Pierre Joseph Louis Fatou. This work was so important that it earned him the Grand Prix of the French Academy of Sciences and made him famous in almost all the important mathematical centers of his time (the Academy also recognized Fatou's contribution by awarding him a second prize).
In 1919, Hausdorff formulated the concept of dimension.
In 1961, Lewis Fry Richardson, in one of his posthumous articles, posed the question of how to measure the length of a border or irregular coastline, as part of his studies on the causes of war between two countries with a common border. He observed that there were great differences in measurements, which varied according to the unit of measurement.
In 1967, Seymour Paper, Daniel Bobrow and Wally Feurzeig invented the Logo language for teaching purposes. This language uses the so-called "turtle geometry", a way of generating geometric objects locally, as an alternative to coordinate-based graphics systems. The turtle moves leaving a trace (a line) by means of commands such as move forward so many steps, turn a certain angle, etc. and using recursion.
In 1968, Aristid Lindemayer, invented the L-system, a language for simulating the growth of living organisms, inspired by the Logo language. For example, the Koch curve in L-system is:
Alphabet: F + − (F indicates segment, + indicates 60º rotation, − indicates −60º rotation)
Axiom: F
Rule: F → F + F −− F + F
In the 1970s, with the development of computer graphics, these fascinating structures including self-semanities began to be studied, mainly by Benoît Mandelbrot, who sensed their enormous potential to build models of reality. Mandelbrot called these forms "fractals" (from the Latin adjective fractus, interrupted, broken, irregular). "Fractal" has the same root as "fragmented" and "fractional". This name has been very successful and has contributed to its rapid diffusion. According to Mandelbrot, "fractal" is etymologically opposed to "algebra", since "algebra" (from Arabic, jabara) means to unite, to bind, and "fractal" is to disunite, to break, to fractionate.
Mandelbrot is considered the father of the fractal revolution. "Clouds are not spherical, nor mountains conical, a coast is not a circumference, nor a crust a plane, and neither does lightning trace straight lines." "I have conceived, fine-tuned and extensively used a new gemmetry of nature" (Mandelbrot).
The founding act of fractal objects as we know them today is his book "The Fractal Objects" [1987]. "My book [...] is a historical document" (Mandelbrot). Subsequently, Mandelbrot published a much more extensive and complete book: "The Fractal Geometry of Nature" [1997]. In these books, reference is made to Richardson in a chapter entitled "What is the length of the Brittany Coast?". He also acknowledges that the idea of recursive self-similarity comes from Leibniz.
In 1981, J.E. Hutchinson [1981] developed a unified theory of self-similar fractal sets by contractive similarity transformations. A transformation is contractive if every pair of points is closer together after such a transformation.
A.R. Smith (in 1984) and P. Prusinkiewicz (in 1986) represent L-systems by computer, producing near-real-looking modelizations of plants, trees and shrubs. L-systems make it possible to generate classical fractal sets in a simple manner.
In 1985, Michael F. Barnsley generalized Hutchinson's method using Iterated Function Systems, IFS. An iterated function is a finite set of contractive transformations of the type W(x, < i>y) = (ax+by+e, cx+dy+f).
This type of transformation allows to perform rotations, translations, symmetries and homotopies (scale changes in X and in Y) to any set of the plane.
The procedure for generating a fractal consists of taking any point and applying one of the Wi at random. The point is drawn. Choose another Wi at random and draw the point again. And so on. The advantages of this technique are:
You can generate self-similar fractals or not.
Fractals can be generated that can approximate, as closely as desired, any natural image, no matter how irregular.
Simply store the family of contractions to regenerate the image when desired, thus saving memory.
Allows to generate the classical regular sets of Euclidean geometry.
Barnsley justifies the IFS method by means of the "Collage Theorem", published in his book "Fractals Everywhere" [1988], which allows to approximate by fractals an image, decomposing it into parts, in such a way that each of them is obtained from the total image by a contractive application (it keeps the shape and changes the size).
The Mandelbrot fractal
Also called the "Mandelbrot set", and also "the Buddha", it is probably the most famous fractal and one of the great mathematical discoveries of the 20th century.
Mandelbrot fractal
It is a figure of enormous complexity (in fact, it is the most complex form of mathematical shapes). The Mandelbrot set contains infinite copies of itself.
At each point there is a Julia set, the one corresponding to the constant c. The Mandebrot set is the set of points c for which the associated Julia set is connected. These Julia sets are repeated over and over again when zooming, but the Mandelbrot set (the continuous curve) is constantly changing, and represents the transitions between the possible Julia sets. The Mandelbrot set is the union of all possible Julia sets in the complex plane.
Its generation is very simple, since it is generated by the recursive function in the complex plane z = z2 + c. The points of the complex plane that converge (those that reach a finite value) are represented, that is, the border points between convergence and divergence. The result is a single continuous curve.
It is the paradigm of the union of the simple and the complex. "The whole is of a staggering combination of extreme simplicity and dizzying complexity" (Mandelbrot).
Paradoxically, Mandelbrot's fractal −the most famous fractal of all− is not really a fractal, because it lacks the property of self-similarity. By studying its boundaries in finer and finer detail, the same structures are not repeated, but always new geometries appear. This fractal includes all the fundamental properties of nonlinear functions.
Mandelbrot's set, apart from its unusual shape and enigmatic beauty, is of great theoretical and practical importance:
It is the maximum curve that fills the 2-dimensional space. In fact, its dimension is 2.
It is related to the double-period chaos equation. A path to chaos is a mechanism by which a dynamical system with one parameter goes from a non-chaotic state to a deterministic chaotic state (a function of that parameter). The most famous route to chaos is a double-period cascade, which serves to describe the expansion of a population, the growth of plants, the instability of atmospheric weather, and other physical processes.
It has many technological applications: data storage, information analysis, fractal antennas, etc.
Finally, the Mandelbrot set has been proposed as a symbol of consciousness and also of the collective unconscious.
Fractal dimension
The fractal dimension D (or Haussdorf-Besicovitch dimension) is different from topological dimension DT and is defined as
D = lim(log N(h) / log 1/h)
when h → ∞
where N(h) is the number of elementary objects of magnitude h that are needed to cover the fractal set. The dimension of a segment and that of a circle is 1. For a regular object formed by equal elementary objects with perfect self-similarity, the fractal dimension is D = log(N(L))/log(1/L), where L is the length of an elementary object and N(L) is the number of such elementary objects.
In the case of the Koch fractal, L=1/3, N=4, D=log 4/log 3 ≅ 1.1618 and DT=1 (as a line that is).
In the case of Cantor powder, L=1/3, N=2, D=log 2/log 3 ≅ 0.6309 and DT=0 (as geometric points, even though they are, in theory, infinitesimal).
The dimension of the Peano and Hilbert curves is 2, since they cover all the points of a square.
According to Mandelbrot, a more general definition of fractal is "a set whose fractal dimension D is greater than its topological dimension DT". This definition includes objects that may not be self-similar.
Fractals vs. holograms
It is often said that the universe is holographic, where each part reflects the whole. However, it is more accurate to speak of the fractal universe. The differences are as follows:
In the fractal and in the hologram there is an all-part relationship. In the fractal the same pattern is repeated at all levels. In the hologram each part of the image (however small) contains the whole image. Therefore, holograms have two levels and fractals have infinite ones.
In the fractal there is self-similarity. In the hologram there is identity.
In fractals the generating laws are simple and complexity is easily generated by recursively applying these same laws or rules. The degree of complexity depends on the number of levels and the generating laws. In the hologram the complexity is always the same.
In the fractal the generating laws are different in each particular fractal. Therefore, in fractals there is freedom of creation, without determinism, which allows infinite diversity. In the hologram the procedure is always the same; in the hologram there is no freedom.
Holograms evoke the third (spatial) dimension. Fractals evoke infinite dimensions.
It is often said that the universe is holographic. It is truer to say that the universe is fractal. All physical reality, from the smallest subatomic particle to clusters of galaxies, seems to follow fractal-like laws, which must necessarily be simple but which manifest themselves with great complexity in their unfolding.
The logarithmic spiral and the golden ratio are fractal
The logarithmic spiral, also called the equiangular spiral or "spira mirabilis" (as Jackob Bernouilli called it), can be considered a fractal because it includes, repeats or self-regenerates itself at all scales. It is the most common spiral in nature (sunflowers, pineapples, roses, nautilus, snails, arms of spiral galaxies, squalls, etc.).
Logarithmic Spiral
There is a very close relationship between the logarithmic spiral and the golden ratio. The golden ratio Φ is the most primary and archetypal manifestation of fractal structure because it preserves its relationship to itself. The main characteristic of a fractal is its self-similarity at all scales.
The equation that defines the golden ratio is Φ = 1 + 1/Φ. Therefore, the golden ratio is a fractal expression, since it includes itself at all levels. The golden ratio is the simplest fractal and the most economical, since it refers to itself. Natural systems tend towards a state of maximum economy and maximum simplicity, so the golden ratio is the most used fractal in nature.
Relativity theory of scale
Proposed by Laurent Nottale [1997], the theory of scale relativity is a unifying theory of the microscopic and the macroscopic, of quantum mechanics and relativity theory, of classical physics and modern physics.
In the theory of relativity the laws of nature are the same for all reference coordinate systems. The theory of relativity of scale adds to this by saying that these laws are also the same whatever the scale of the coordinate system, thus generalizing the principle of relativity. According to this theory:
All physical quantities are of fractal type, including space-time. Physical quantities are functions, because they depend explicitly on the resolution, that is, on the precision of the observation. Moreover, they are never exact, they are always approximate. For example, a steel bar, when observed closely, will present pores, so that it cannot really be considered as a three-dimensional object, but an object of fractional dimension (between 2 and 3), i.e. a fractal.
Quantum and classical behaviors are a matter of scale. At the microscopic scale quantum properties appear. At the macroscopic scale the properties are classical. The different observations in the micro and macro worlds do not come from the laws being different, but are the same laws manifesting themselves at different scales.
Space-time manifests itself as fractal when observed at the subatomic scale. We can visualize space-time as a "foam".
The fractal property of space-time manifests itself in the irregular trajectories of particles (a property discovered by Feynman), which are not differentiable. According to Lebesgue's theorem, every continuous non-differentiable curve has infinite length. Therefore, the length of each trajectory is infinite (like every fractal curve). That is why it is not observable and has been interpreted probabilistically.
A trajectory of a free particle is a geodesic of space-time. At small scales there are an infinite number of geodesics. A particular geodesic is associated with the corpuscular nature of a particle.
The fractal/non-fractal transition corresponds to the quantum physics/classical physics transition. The De Broglie wavelength, associated with a particle, is the threshold between fractal (quantum mechanics) and non-fractal (classical mechanics) behavior.
In the same way that the curvature of space-time defines gravitation, fractal space-time defines quantum properties. The structure of space-time is both curved and fractal (non-continuous).
In the same way that at large scale there exists an invariant of maximum type (the speed of light), there exists at small scale an invariant of minimum type, not surmountable, which is the so-called "Planck length" (1.6*10−35 m).
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