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Fractals - fascinating mathematical objects
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Fractals - fascinating mathematical objects
Fractals are great tools for understanding the world and other complex structures. We use them to analyze financial processes, biology, computer science and mathematics, image compression, life sciences and technologies, and many other disciplines.
Fractals: definition
A fractal figure is a mathematical object that has a similar structure at all scales.
It is an "infinitely fragmented" geometric object whose details are observable at an arbitrarily chosen scale. By zooming in on a part of the figure, it is possible to find the whole figure; it is then said to be " self-similar ".
Fractals are defined in a paradoxical way, a bit like Russian dolls that contain a figure that is more or less identical to the scale: fractal objects can be considered as nested structures at any point - and not only at a certain number of points. This conception of fractals implies this recursive definition: a fractal object is an object of which each element is also a fractal (similar) object.
They often appear in the study of chaotic systems.
In mathematics, and in a simpler way, these geometrical curiosities are figures whose structure is invariant by change of scale. Thus, if you zoom in on a part of a fractal, you will find a structure that reproduces itself again and again, smaller and smaller... until infinity. Similar details can be found at arbitrarily small or large scales.
Who invented the word fractal? Mandelbrot's fractals
Many examples of fractals, such as the Koch flake or the Serpiński carpet were discovered in the late 19th century, but it was the mathematician Benoît Mandelbrot who in 1975 drew attention to these objects and their ubiquity in nature, creating the adjective "fractal" from the Latin root fractus, which means "broken", "irregular", and the inflection "-al" present in the adjectives "naval" and "banal" (plurals: navals, banals, fractals); usage later imposed the noun a fractal to refer to a figure or equation of fractal geometry.
Benoît Mandelbrot defines fractals as objects that do not lose their details or proportions if they are enlarged or shrunk, even on a microscopic scale. This property is reminiscent of phi, the golden ratio, 1.618, for which the same sacred and essential proportion is preserved at each section of the line or rectangle
In fact, the properties of phi and fractals have to do with growth.
Where can I find fractals?
There are two types of fractals, geometric fractals and natural or random fractals.
Natural fractal structures in nature
The structure of a flake is one of the clearest manifestations of fractals in nature. This may be due to the fact that the flake is formed when water falls freely from the sky, passing through the atmosphere without encountering any interference. No other matter crystallizes in so many forms.
Despite this variety, the geometry that governs the growth of one of the branches of the flake will also govern the growth of the other branches. A kind of geometric coordination is at work. Regardless of the scale used to observe the finished product, the pattern is identical
However, these figures do not have the clarity of Euclidean geometry, where everything is either straight, circular, or a curve generated along a cone.
The Koch flake
In 1904, the mathematician Helge Von Koch (1870-1924), developed a mathematical model for producing a single flake pattern. He started with a simple equilateral triangle to generate the curve of this flake.
1) First draw an equilateral triangle.
2) Overlay a second inverted equilateral triangle to form a hexagram.
3) Each vertex is an equilateral triangle pointing outward, overlaid with an inverted equilateral triangle.
4) Delete the hexagram lines remaining in the original triangle.
5) Repeat this process for each point of the original hexagram.
6) Repeat in a fractal fashion, to an increasingly fine level of detail.
The procedure that allows to generate a Koch fractal representing a snowflake is perhaps not the reflection of what happens in nature on a cold day, but it allows to obtain a mathematical representation of the fractal nature of the snowflake.
It also demonstrates that if man were endowed with supernatural vision, he could, with an extra-fine pencil, repeat the figure until reaching the infinitely small.
Plants
Approximate fractals are easily observed in nature. These shapes have a self-similar structure on a large but finite scale: clouds, snowflakes, mountains, river systems, 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 evident in these examples - the branch of a tree or the frond of a fern are miniature replicas of the whole: not identical, but similar in nature.
Logarithmic spirals
The French mathematician and philosopher René Descartes (1596-1650) was the first to describe the spiral, which is now called a logarithmic spiral. However, it was the Swiss mathematician Jacob Bernoulli (1654-1705), fascinated by its extraordinary mathematical properties, who called it spira mirabilis, "marvelous spiral" in Latin.
As the size of this spiral increases, its shape remains the same, as it expands at a constant rate in a geometric progression. These beautiful spirals, also called equiangles or exponential spirals, are found everywhere in nature, in living creatures, in galactic hurricanes and other natural phenomena.
The nautilus shell has some of the most beautiful, graceful and recognizable spirals in nature.
The nautilus, while passing from one compartment to another, larger, fills the previous one of gas and closes it completely of a mother-of-pearl plug. It occupies only the last of the lodges, but leaves behind it a tiny thread which winds up to the lodge of origin. Each additional compartment is proportional to the previous one, a geometrical figure whose angle formed with the center of origin is maintained with the wire of growth.
The shell and the Mandelbrot spiral are in fact derived from the Fibonacci spiral, also called the golden spiral.
Clouds
As surprising as it may seem, the cloud, which is formed by condensation of water vapor, has a broken and fragmented outline which has the same pattern when we change scale, it has a surface with a fractal curvature.
Like water, air takes on the shape of a spiral during tornadoes or storms; we see this clearly in the image below.
Fractals in the universe
In cosmology, the fractal universe model designates a cosmological model whose structure and distribution of matter have a fractal dimension, and this, at several levels. More generally, it corresponds to the use or appearance of fractals in the study of the Universe and the matter that composes it.
The first snippets on the theory of a fractal universe are born with the mathematician Benoît Mandelbrot. In his book "Fractal Objects: Form, Chance and Dimension" published in 1977, he mentions that galaxies have a fractal distribution and outlines the properties of such a distribution.
Subsequently, in 1986, the Russian theoretical physicist Andrei Linde wrote the article Eternally Existing Self-Reproducing Chaotic Inflationary Universe published in the journal Physica Scripta where he uses fractals to explain his vision of the universe. The following year, the Italian professor Luciano Pietronero published a first model of galaxies according to a fractal distribution in an article published in the journal Physica A.
Fractals in mathematics: geometry, triangle and other figures
In mathematics, a fractal is a metric space whose Hausdorff dimension (denoted δ) is strictly greater than the topological dimension. This is at least the definition initially given by Benoît Mandelbrot, but he quickly replaced it by a more vague definition, allowing to include for example the Hilbert curve.
Some shapes have a more or less important complexity.
The Hausdorff dimension of a metric space (X,d) is a positive or zero real number, possibly infinity.
The Hilbert curve is a continuous curve filling a square.
Let's see some examples of fractals in the field of mathematics.
Sierpiński's carpet
The Sierpiński rug (created in 1916), named after Wacław Sierpiński, is a fractal obtained from a square. The mat is made by cutting the square into nine equal squares with a three-by-three grid, and removing the center piece, and applying this procedure indefinitely to the remaining eight squares.
Sierpiński's triangle
The Sierpiński triangle, or Sierpińsky sieve, also called by Mandelbrot the Sierpiński breech joint, is a fractal, named after Wacław Sierpiński who described it in 1915.
The Sierpiński triangle can be obtained from a "full" triangle, by an infinite number of repetitions consisting of dividing the size of the triangle by two and then joining them in triplicate by their vertices to form a new triangle. At each repetition the triangle is thus of the same size, but "less and less full".
Construction of the Sierpiński triangle:
Menger's sponge
The Menger sponge, sometimes called the Menger-Sierpinski sponge, is a fractal solid. It is the extension in a third dimension of the Cantor set and the Sierpiński mat. It was first described by the Austrian mathematician Karl Menger (Menger 1926).
This figure, which comes straight from Karl Menger's imagination, resembles a sponge whose shape is that of a cube pierced by a multitude of pores all connected to each other. Menger wanted to prove that it was possible to obtain an infinite surface in a finite volume.
Construction: If we divide each of the edges into 3 equal parts, each face will be formed by a checkerboard of nine squares. Let's start by emptying the middle one. By adding the walls of this hollowed out part, the surface area of the structure is then larger than that of the original cube. Thus, we increase the surface area without changing the volume.
Each of the 8 remaining squares is now divided into a tiny checkerboard of 9, whose central figure is again hollowed out... and so on, until microscopic portions are reached. By digging into the starting volume, the surface keeps increasing, admittedly by a smaller and smaller amount, but... without any limit. In the end, we will have a three-dimensional lace which will not overflow the original cube.
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For more information, here is the link to Wikipedia.
Art and fractal design : when mathematics rhymes with beauty
Fractal art represents the alliance of art and mathematics. This recent art form uses computers to make images from mathematical formulas. Fractal artworks are often geometric in appearance, with intricate patterns and a wealth of detail.
Fractal art is a form of algorithmic art that consists in producing images, animations and even music from fractal objects. Fractal art developed from the mid-1980s.
It is a kind of digital art.
Indeed, fractal art is rarely drawn or painted by hand, but rather created with the help of computers, which are indeed able to calculate fractal functions and generate images from them. It is in fact the appearance of computers that has allowed the development of this art, because it requires a lot of computing power.
The increase in computer power has allowed the creation of software allowing the calculation of three-dimensional images in computer graphics, thus offering the functions and effects usually reserved for classical three-dimensional modeling software (lights, volumetric lights, depth blur, atmosphere, reflection/refraction of certain materials, textures, ...).
The artistic process for creating a 3D fractal is the same as for a 2D fractal. The fact that Kerry Mitchell (an American artist known for his algorithmic and fractal art, which has been exhibited at the Nature in Art Museum, The Bridges Conference and the Los Angeles Center for Digital Art, and for his "Fractal Art Manifesto") wrote that fractal art was"a subgroup of two-dimensional visual art" does not mean that the realizations made in 3D are not fractal art but that 3D fractals simply did not exist yet in 1999 because the power of computers at that time did not allow it and nobody had yet produced a three-dimensional version of the Mandelbrot set.
Where can you find fractal art?
Well, you should know that all our Mandalas are fractal Mandalas!
All our Mandalas are in 3D. That's why you often have the impression that they move! But it's also because they are alive since they were created by connecting to the wisdom of living nature.
Their complexity is such that they are incomparable, you will not find similar ones anywhere else!
This is also the richness of our store because we have at our disposal more than 400 highly vibratory Mandalas that are at your disposal to accompany you in your daily life. Each one radiates an energy of its own
How to make or draw fractals?
As you can see, drawing fractal art by hand is very complex and seems to be an attempt that might fail.
Fractal generator software is any type of graphics software that generates images of fractals. There are many fractal generator programs available, both free and commercial. Do your research on Google!
To conclude
There are a wide variety of examples of fractals, from fractal landscapes to the alveoli of the lung. Characterized by three-dimensional space and self-similarity properties, fractals are all around you and some of them will definitely not leave you indifferent!
This is the case of our vibratory Mandalas which today are spreading around the globe.
Feel free to browse the store to discover their incredible richness and beauty.
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And if you want to go further in the discovery of symbols, welcome to this space / store dedicated to sacred geometry. You will find a multitude of fractal mandalas.
Sources :
Géométrie sacrée (Éditions Véga)
The Golden Number (Éditions Dervy)