Fractals in Nature

The Telnet Redes Inteligentes ENVIA antennas are characterized by their small size and the ability to host multiple operators in a single antenna. This is possible due to its design is not the same as the traditional antennas, but are based on fractal geometry.

Fractal design of antennas allows much smaller elements to be used to transmit the signal and admit frequency ranges much wider than a traditional antenna. Moreover, fractal geometry allows to combine antennas of different frequencies avoiding interference, allowing us to design tri-band and tetra-band antennas with a very small size and the same gain factor as a traditional antenna. To get an idea of the degree of integration, a Tetra-Band ENVIA Compact Antenna is able to replace within a single radome mast and antenna up to 12 traditional single-panel antennas.

However, fractal geometry is not something that was designed exclusively for telecommunications, but has always existed in nature and are discoverable to the naked eye.

Fractal Forms

In order to identify fractal forms in nature, we must first know what a fractal is and its properties.

Benoît B. Mandelbrot, was who first gave the name "fractal" to this type of geometry and defined them as a semi-geometric object whose basic structure, fragmented, or irregular, is repeated at different scales (Mandelbrot, The fractal geometry of nature, 1982).

K.J. Falconer defined the following properties for fractal forms (Falconer, 2003):
•    Too irregular to be described in traditional Euclidean geometric terms.
•    It has detail at every scale of observation. This property implies that in times when fractalisation, or the degree of detail of the geometry, tends to infinity, it is impossible to measure. This has made that in the mathematical literature to be called monster curves, since the length of a fractal curve iterated ad infinitum, contained in a finite area, is infinite, something that in traditional geometry is itself a paradox.
•    Its Hausdorff-Besicovitch dimension is strictly greater than its topological dimension. This theoretical property means that its size, either in length or area, depending on which has the fractal dimension, is greater than would be expected in an euclidean form. The Hausdorff-Besicovitch dimension is a measure of the fractal dimension or the "fragmentation" of geometry to study and gives us an idea of how it occupies the space that contains it.
•    It has self-similarity, either exact, quasi self-similarity or statistical caused by the following property.
It is created from a recursive method. Such methods are those that repeat the same rule applied successively to the previous result. This is the main reason why we find fractals in nature so abundantly, as occurs in geologic processes (rain erodes the same mountain that has eroded a rain the previous day) or the growth of many biological structures, which use this type of resource for development of complex structures.

But not all fractals are equal. So a new term is introduced to characterize: the fractal dimension. This measure can be measured in different ways, and what he intends is to give an idea of how occupies the space in which it is contained the fractal shape. The more fractal dimension has, the more rapidly grow while being scaled and therefore more length, area or volume occupies within the Euclidean dimension that contains it (if a curve, area or if a surface, a volume).
 

It is called fractal dimension because, unlike traditional euclidean objects (a curve has dimension 1, the surface of a plane has dimension 2 and the volume of a sphere has dimension 3), however, a fractal curve can have a dimension 1,26 (Koch curve) or the fractal dimension of broccoli's surface is 2.66, for example.

Fractals in Nature

Mandelbrot begins the introduction to his book "Fractal Geometry in Nature" (1982) as follows: "Clouds are not spheres, mountains are not cones, coastlines are not circular, and barking are not smooth, as lightning does not travel in straight lines." (Mandelbrot, The fractal geometry of nature, 1982).

The characteristics of fractal forms are given by their own process of creation and can be found in different aspects of nature, as for example in the field and geology. In 1967 Mandelbrot published an article in Science magazine titled "How Long Is the Coast of England? Statistical self-similarity and fractional dimension". (Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, 1967). This article explains the "paradox of the coast". This is based on the length of the coast of England varying depending on the length of the ruler used to measure it. This is because the coastline is a type of fractal generated by erosion and the composition of the rock. This creates a multitude of inbound, outbound, which are, at large, curves with fractal properties.

In turn, the mountains also have fractal geometry, in this case caused by rain erosion, wind, rocks fractured by changes in temperature and pressure and earthquakes that create the rocky ridge on which sits the mountain. These landscapes can even be parameterized and generated virtually with digital imaging software, using self-similarity and recursive algorithms.

The mouths of some rivers also exhibit this type of geometry, caused by the branching of different flows. In these cases, as may also happen in the branches of trees or lightning, there is no exact self-similarity, but a quasi-similarity or statistical self-similarity.

The cause why these such different phenomena in principle all have fractal properties is because they share the same creation process, called diffusion-limited aggregation or fractal growth (Sander, 1987). This type of growth defines the way in which the cost-lines are created, the spread of the rays or the growth of blood vessels in our body (Ary L. Goldberger, 1990).

Fractals in Livings

Surprising examples of fractal geometry can also be found among living things. One of the most striking one is the Romescu Broccoli, which has fractal geometry with an almost perfect self-similarity. Also regular broccoli present fractal branching, although the self-similarity in this case is not as accurate.

One of the most classic examples of fractal geometry in nature comes from one of the first plants on our planet: the ferns. These have an almost perfect self-similarity between its branches. The reason for the appearance of this type of fractal forms in living organisms is due, as in the case of coasts and mountains, which uses a method of creating simple and repetitive to generate complex shapes.

Ramification is one of the most abundant biological design due to its simplicity and efficiency in covering a surface or volume, a property that the ramifications share with the rest of fractals while being a "monster curve" of limited iterations. The plant's genetic code gives the same order to the main branch and to a secondary one: growing and split creating a copy of yourself in each branching. Thus, we find branches with fractal shapes in both ferns and trees, the leaves of the same or even in our own nervous system, cardiovascular or alveoli in our lungs.

In the case of plants, this design allows them to maximize the surface and thus capture more light, CO2 and oxygen as possible. In the case of the nervous system or veins and arteries of our body, they can reach and feed the maximum number of cells and ensures that blood pressure per cm2 in each of the branches is the same while having self-similarity and its fractal formation process, like the fractal shape of our bronchi and pulmonary alveoli allows us to maximize the exchange of CO2 and oxygen in each breath. To get an idea of the level of branches and the great use of space to get into the lungs with this fractal design it is enough to know that the fractal dimension of the inner surface of the lungs, measured from casts of human lungs and other species is 2.7, when a traditional Euclidean plane has dimension 2.

But not only the shapes of our bodies are fractals, but if we study the body functions we can also find fractal patterns in them, and even this fractal property can be a sign of health as opposed to regular cyclical patterns (Ary L. Goldberger, 1990). It has long been considered in traditional medicine that regular heart rate was a sign of health and when aging body rhythms appear chaotic and erratic as a sign of disease. However, recent studies have shown that heart rates over time has a fractal shape and at first seems chaotic, in contrast, repetitive and periodic patterns are a sign of disease. This is because a healthy heart is able to change your heart rate to compensate for the needs of the organism, transmitted by the sympathetic and parasympathetic systems, creating these chaotic oscillations. A diseased heart is not able to adapt and meet the needs of the organism, and presents a regular pattern, which eventually degenerate tissue, producing a system failure.

Also has been associated with fractal structures in physiological systems of distribution (blood system, lymphatic), collection (digestive, pulmonary) and information processing (neurons and nervous system) with resistance to injury and partial failures, because of its self-similarity and redundancy of structures (Ary L. Goldberger, 1990). This means they can function in case of illness, trauma or injury caused by stress and aging, allowing healthy areas to supplant the functions of the damaged ones.

Bibliography

Ary L. Goldberger, D. R. (1990). Caos y fractales en la fisiología humana. Investigación y ciencia , 31-38.Falconer, K. J. (2003). Fractal geometry: mathematical foundations and applications. Wiley.Mandelbrot, B. (1967). How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science , 636-638.Mandelbrot, B. (1982). The fractal geometry of nature. W.H. Freeman.Sander, L. M. (1987). Crecimiento Fractal. Investigación y Ciencia , 66-73.