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Geometry of the nature – Fractals

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Mandelbrot, computer graphic, and fractals

Fractal structures as a portrayal of the natural growth

Look closer at the nature and almost everywhere you will find fractals. For example, a simple tree: what you see, when you look at it?

Silhouette eines Baumes
Example of a tree

Look closer at the nature and almost everywhere you will find fractals. For example, a simple tree: what you see, when you look at it? A trunk with a set of branches in a random order? The whole object seems as random one, created without any rules and templates.

If we don’t think about fractals! The situation gets different, as soon as we try to apply this special geometric knowledge to what we see around us. Then we notice, that a tree, actually, is composed from many other, smaller trees, which branch off it, and which, in their turn, also are sets of smaller trees, and so long, and so further. This simple principle is a basis for all trees and the majority of plants.

„Clouds are not balls, mountains are not cones, coastlines are not circles and tree barks are not smooth, as well as a thunder bolt follows not a line.“

Benoît Mandelbrot

 

Design from the nature

farn with fractal structure
farn with fractal structure

One of the almost perfect fractal objects is a simple fern. It consists of a stem and many smaller ferns on the left and the right of it. These, if to look closer, also bring many small ferns on both sides of the little stems and so forth. A further typical example from biological ones is a fractal based scheme of a Romanesco broccoli. Even a cauliflower has fractals in its design, although we might not notice it from the first glance.

So we could say that growth in the nature follows the fractal laws, when objects copy themselves into smaller ones, and again, and again. Such copies are called satellites. On the rim of such satellite we see almost the same structures like on the corresponding places of the big original object.

The situation is similar to one, which can be found in any living organism and its genes. Each satellite corresponds to genetic substance of a cell, which contains the construction plan of the whole organism, though from the beginning only the mother structure is obviously visible to us.

Mountains
Mountains
Clouds
Clouds
Snowflakes
Snowflakes

 

 

 

 

 

 

Observing more detailed, fascinatingly, we find fractals also in coastlines, mountains, cloudscapes, blood circulatory system, system of the rivers, snowflakes, crystals, spread of stars and more other. Although, we aren’t speaking here about 100% copies, seeing the natural growth as fractals opens doors to the better understanding and possibilities of calculation and illustrating these processes. This allows wonderful results even without precisely following the similarities.

Plants
Plants
System of the rivers
System of the rivers
Blood circulatory system
Blood circulatory system

 

 

 

 

Thunder bolt
Thunder bolt
Treetops
Treetops
Stars dispersion
Stars dispersion

 

 

 

 

 

Legacy of Benoît Mandelbrot

Benoît Mandelbrot
Benoît Mandelbrot

Often you can hear fractals being named Mandelbrot. Famous mathematician Benoît Mandelbrot was the one, who formulated the definition of fractals. As a fractal understand man a geometric template, which has such characteristics like broken dimensionality and a high level of self-similarity. Our example, when an object consists from a set of smaller copies of itself, illustrates it well.

As the name implies, is the classical definition from the Euclidean geometry extended, reflecting the presence of broken and not natural dimensions of many fractals. Two other mathematicians, Wacław Sierpiński and Gaston Maurice Julia, belong near to Benoît Mandelbrot to those, who investigated fractals and gave them their name.

In contrast to the Euclidean geometry, which makes objects flatter and, therefore, simpler (for example, circle), enlarged fractals show more complex and new details.

Fractal computer graphic
Fractal computer graphic

Fractal computer graphic

Rich forms of fractal borders are especially attractive from the graphic point. The more we enlarge a fractal, the more complex structures can we observe. These can be studied and illustrated like under a microscope with a help of the corresponding computer programs. In this case, there are only two artistic freedoms are left: a choice of the detail to repeat and colors of the image.

Universe of fractals
Universe of fractals

A universe in a proton on the left of the nutshell

In order to investigate the most interesting structures we often need such enlargements, which are impossible to produce with the normal accuracy of calculations of the popular program languages. High level of calculation mistakes and rounding errors are the main reasons of it.

However some special software allows working with arithmetic operations, which require 100 or far more decimal places, and gives an opportunity to make significant enlargements. Such magnification factor can reach a level of 10100 and more, what gives almost astronomical results.

Fractal computer graphic
Fractal computer graphic

To compare with: a magnification scale of a proton to sizes of our universe is 1040. With 10100 we step over the limits of imaginable.

The Mandelbrot’s set is a rich of forms, geometric structure. It inspires lots of designers, filmmakers, and computer artists. Even some couturier use shapes of complex fractals to create their best new models.

 

From the moment I first got in touch with the fractal geometry I was many times deeply inspired, discovering these forms in the nature and realizing more about its fascinating and unique design. With the help of Photoshop this knowledge can be wonderfully put into practice letting create sets of realistic and difficult compositions. Not to mention those opportunities, which lay in 3D environments.

Abgelegt in: Grafik Design Datum: 5. Februar 2013 Erstellt von Stephan Bender

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Dok: Ben.design Grafik DesignGeometry of the nature – Fractals