Fractals are much more than cauliflower

A cauliflower or a broccoli. They are the most popular examples of fractal systems. A tree too. After all, these are easily recognizable examples made up of structures that are repeated and are exactly the same as each other.

Any branch of a tree follows a structure similar to the others, and multiplies incessantly until it forms the tree itself. But we warn in the title that fractals are much more than that, they are a code that exists in complex systems and knowing them means being able to decipher how they are going to behave. They are often referred to as “the geometric code of nature”.

In the second half of the 20th century, a group of mathematicians and physicists proposed new models to study these complex and chaotic phenomena. These models are called fractal sets, and are based on the concept of self-similarity, that is, by enlarging a part of the object, a pattern similar to the complete design (the tree, the cauliflower, etc.) is observed.

What do we mean by complex systems? There are examples of fractal sets that are less intuitive than a tree, such as the turbulence of a storm, the way lightning propagates, the circulatory or neural system, the relief of a landscape, and also the stock market, rising prices, etc. even the evolution of a pandemic like covid-19. We are facing a new method to understand the world.

Predict the evolution of covid-19 with fractals

How will covid-19 evolve? Answering this question is among the great scientific priorities of the moment. Traditional techniques only allow us to analyze how the expansion of a pandemic has been once it has ended. However, the spread of a pandemic, a chaotic and complex system, can be seen as a fractal system. In its development there are patterns, a code, tree branches that are repeated. Knowing them, it is possible to predict how it will evolve even if all the data is not available.

With the data recorded in a certain time interval, the fractal interpolation can estimate the intermediate data between the real ones, comparing similarities (the branches of the tree) with the perspective of the structure of the initial data.

In a study published in Chaos, Solitons & Fractals, an analysis of epidemiological curves of covid-19 corresponding to five European countries is presented after approximately one hundred days of registered cases. It is verified that the data has a fractal structure, a characteristic that can provide a different perspective to approach the analysis of the incidence curves of the disease.

How long is the coastline of Great Britain?

One of the pioneers in proposing mathematical procedures for the analysis of irregular phenomena was Professor Benoit Mandelbrot, father of the fractal theory, whose applications are still being developed today. A Mandelbrot phrase that has gone down in history: “Clouds are not spheres, mountains are not cones, and lightning does not travel in straight lines.”. In this theory it is accepted that the dimension of an object can be fractional, for example, 4/3.

One of Benoît Mandelbrot’s most famous articles is entitled How long is the coastline of Great Britain?. In it he takes up a work by the brilliant British scientist Lewis Fry Richardson who, in 1961, conducted a study to measure the length of this coastline. An approximate way to make this measurement is to consider a given step length (h), and estimate the number of steps needed to complete the route.

With this method, the product of both quantities gives an empirical length L(h). But he has a catch. Richardson observed that as the step h gets smaller, the magnitude of the coast increases indefinitely. In mathematical language we would say that the length is infinite, and with this Mandelbrot’s question was resolved.

But Richardson had something else in mind, and that is that Britain’s coastline is not a perfect plain. He discovered, for mathematics, that the length L(h) is proportional to h raised to -α. This exponent α is the one that quantifies, the one that takes into account, the irregularity of the coast. This constant is a precedent for what is now called the fractal dimension.

Mental maps

The study of bioelectrical signals such as those produced by the human brain was classically carried out by means of frequency analysis, that is, the number of cycles per second described by a wave was computed. However, the pre-20th century models they are great whenever things behave wellthat is, they are not complex or chaotic.

Fractal curves present geometrically complex graphs that are constructed from a series of initial data. Using this type of techniques, a fractal dimension of the record can be obtained. This indicator can be used to make brain maps and be able to compare different mental states corresponding to groups of healthy patients and others affected by some type of disorder. Mental illness.

Using fractal parameters, differences in brain activation can also be detected by the exercise of certain attention testwith the consequent knowledge of the brain areas involved in each of them.

The above examples (there are many more) show that we are faced with a new perspective to address the great paradigm of contemporary science: understanding a complex reality, changing even, sometimes, hostile.

Font: María Antonia Navascués Sanagustín / THE CONVERSATION

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