In the early 2000s, a theory by the physicist Luciano Pietronero, of the Institute of Complex Systems of the La Sapienza University of Rome, hypothesized that the distribution of matter in galaxies was not continuous, but followed the so-called fractal geometry. The first mapping of dark matter - the hypothetical component of matter that would not emit electromagnetic radiation - carried out by Richard Massey of the California Institute of Technology and published in the scientific journal Nature, seemed to prove the Italian physicist right. “Dark matter is nothing more than a patch that justifies theories – Pietronero argued – fractals clarify these phenomena much better”.
Fractals? But what are we talking about? Let's explain: a fractal is a very particular mathematical object, with astonishing properties. The first is not having one size whole, but fractional. The second is to resemble itself at every scale of representation, even infinitely small (internal omottia). Paradoxes? No, concepts – not abstract at all – which have a lot to do with reality. Just look closely at the fresh snow that settles on the ground, a Roman broccoli (brassica oleracea), a fern, the branches of trees or the lung alveoli. What is missing is perfect regularity and the fact that the magnification cannot continue infinitely, otherwise fractals approximate the shape of these objects much better than traditional geometry.
The father of fractal geometry Benoît Mandelbrot he used to respond in the following way to the question "How much do the coasts of Great Britain measure?": "It depends on the length of the measuring instrument". And, in fact, every time the length of the chosen measuring instrument is reduced, the length of the coast increases, because increasingly smaller sinuosities can be taken into account. Unlike the curves of Euclidean geometry, which become straight lines when scaled up, the fractional folds of coastlines, mountains, and clouds do not disappear when we scale up. If the length of a coastline is measured with increasingly shorter measuring rods, its length grows indefinitely. Instead, think of the tangent of a curve at each of its points: locally, that is, in a sufficiently small neighborhood of that point, the curve can be approximated to its tangent. In the case of a fractal system, there is no regularization of the structure at any order of magnitude, since its complexity and regularity do not vary with scale. This large irregularity cannot therefore be described using traditional mathematical methods.
The first studies on fractals were carried out by people of the caliber of Karl Weierstraß (1872), considered the founder of modern mathematical analysis, followed by the research of the Italian Giuseppe Peano (1890) and, in the twentieth century, by the Polish Wacław Sierpiński and by Frenchman Gaston Julia, as well as Koch and Hausdorff. The aforementioned Mandelbrot coined the name "fractal" only in 1975. The French mathematician was unable to find an answer to the question "How long are the coasts of Great Britain?", but he tried to define a fractional number (between 1 and 2) that would identify the indentation of the coast. Fractal geometry was definitely a “disruptive” topic (breaking with the theories of the past), but should not be seen as a sort of new mathematical order through which the scholars of the time intended to rewrite scientific knowledge. Rather, it was a tool capable of showing how Euclidean geometry failed to represent certain aspects of reality, but at the same time, provided mathematicians with a new yardstick for measuring and exploring nature.
To create a fractal, all you need is a recursive algorithm
In 1906, the Swedish mathematician Helge Von Koch he created a geometric figure with curious characteristics: the length of the perimeter of this object was infinite, but it circumscribed a finite area. Nothing strange for the mathematics of the time, which could boast different types of apparent mathematical paradoxes, resulting from the intuitions of George Cantor. For example, the set that bears his name is constructed recursively starting from a segment along a unit of measurement and removing at each step the central segment 1/3 to 2/3 of that length. The set of points that are never removed by this procedure are as many as the initial interval contains and – it seems incredible, but it is true – it is possible to demonstrate this. How is it possible to demonstrate, through a simple one-to-one correspondence, that the points of a finite semicircle are as many as those of an infinite straight line. The Koch curve is instead obtained by repeating a single instruction (recursive algorithm): an equilateral triangle is taken and a new triangle is built on the third part of each present side, which will have dimensions one third smaller than the previous one. The sides of the triangle thus constructed will initially be 3, then 12, then 48 and so on. The length of the perimeter of this kind of snowflake will be given from 3 x 4/3 x 4/3 x 4/3 up to infinity. Yet the area of the figure thus obtained will remain smaller than the area of the circle circumscribed to the original triangle, ergo a line of infinite length delimited by a finite area!
Mandelbrot in 1977 managed to demonstrate that the curve that outlines Koch's geometric figure is so complicated as to make it impossible to consider it a simple one-dimensional line. But it is not enough to be considered a two-dimensional plane. Using Hausdorff's definition, Mandelbrot calculated the exact size of the Koch curve: log 4 / log 3 = 1,261859507143… and therefore slightly greater than 1. Between 1978 and 1979, Mandelbrot was busy studying a particular mathematical problem: finding the geometric locus of the points of the Gauss plane for which a given sequence of complex numbers was limited, as a function of the starting points. Armed with a computer from Harvard University, he was finally able to create a graphic representation on the monochrome display he had at his disposal. This object resembled a cardioid and was located in the immediate vicinity of the origin of the axes of the complex plane. Observed better in IBM studies with greater resolution and calculation precision, its exceptional characteristics came to light: in particular, the edge of the set proved to be a real mine of surprises, showing an astonishing complexity. At higher magnifications, the cardioid appeared to be made up of dense tangles of bubbles, tentacles, shapes that resembled seahorses, as well as entire small groups scattered almost everywhere. It was therefore interesting to find out whether the islands were islands or connected to the main body by some subtle path of points belonging to the set. The verification came a little later from the mathematicians Adrien Douady and John H. Hubbard, two of the most important figures (Hubbard is still alive) that Chaos theory can include. With modern computers it is possible to analyze considerably larger portions (in terms of infinitesimals) of the plane compared to the first experiments, but an infinite part of the whole will still remain inaccessible as there is an insurmountable limit constituted by the precision of the machine: the number of digits that he can handle. This is one of the reasons why, rightly, the Mandelbrot set is considered themost complex mathematical object of all time.
The use of fractals to describe or model the universe
We explained how fractals are geometric objects that appear the same at any scale, whether you enlarge or reduce them. As astronomers probed ever-larger portions of the cosmos, they were surprised to find matter similarly clumped together, but on ever-larger scales. This distribution of matter, similar to that of a Russian matryoshka, led them to wonder whether the universe is a fractal. One of the very first studies on the hypothesis of distribution of galaxies with a fractal pattern was done by Luciano Pietronero and his team in 1987, while the in-depth study mentioned at the beginning of the article, and which generated the constructive debate among scientists, dates back about a decade later. This is a confirmation of Pietronero's theory itself, made thanks to the availability of a broader cataloging of galaxies. Among the main "adversaries" of this theory was the group of astronomers from New York University led by David Hogg. According to Hogg, the theory of fractals "creates more problems than it solves", starting with the fact that all the fundamental laws of cosmology should be rethought. The debate found fertile ground in the magazine “New Scientist“, where everyone was able to argue their positions, contributing to the enrichment of the community, in terms of new information and connections between them. To date, although doubts remain about "dark matter", the classical theory of the uniform distribution of matter seems to have been confirmed, but it was possible to reach this conclusion precisely thanks to the conduct of the debate at this level. Anyone who is not used to the scenarios of the research world might be surprised by how the event unfolded comparison in these ways, especially having as a yardstick the television diatribes that arise needlessly on many topics related to medical research. Sometimes dissent only exists in mass media or it is generated - in the least suitable places - before scientists can even reach a conclusion, in a more suitable context. From this point of view, it can be noted that the first mapping of dark matter in a part of the universe, which seemed to prove Pietronero right, did not unbalance the Italian physicist, on the contrary: "I would wait before giving definitive sentences", were his words upon publication of the mapping. “More evidence one way or the other will come in the coming months, as new data becomes known.” These were his first utterances.
The first appearance of fractals in cosmology probably occurred with the “theory of the eternal self-reproducing chaotic inflationary universe” proposed by the Russian physicist Andrei Linde, in 1986. When we talk about “inflation” in cosmology, means the hypothesis that the universe, in a very early phase of its existence, went through an extremely rapid expansion, due to a great negative pressure. Several mathematicians and physicists, over the years, have tried to support the theory of a fractal universe, irregular and lumpy, as that of a universe homogeneous and isotropic (uniform) and not only that, because we also tried to reconcile the two models, taking them into consideration on different scales. Modern Big Bang theories predict that our “local” universe began to exist a small fraction of a second after the Big Bang itself, while the rest of the universe continued to expand at an exponential growth rate. The observable part of our universe would therefore only be a particularly hospitable region, a pocket, in which inflation ended and stars and galaxies were born. Globally our universe could be like an infinite fractal, with a mosaic of different pocket universes, separated by an inflating ocean. Local laws of physics and chemistry may differ from one pocket universe to another, which – together – would form a multiverse. For Stephen Hawking, however, this description of eternal inflation, as a Big Bang theory, would be wrong: "the problem with the usual description of eternal inflation - argued the British cosmologist - is that it assumes the existence of a background universe that evolves according to Einstein's theory of general relativity and treats quantum effects as small fluctuations around this. However, the dynamics of eternal inflation erases the separation between classical and quantum physics. As a result, Einstein's theory collapses into eternal inflation. We expect our universe, at the largest scales, to be reasonably smooth and globally finite. So it's not a fractal structure.”
In essence, the universe would be similar to a fractal up to many scales of magnitude, but at some point, this mathematical form collapses. There would be no more Russian matryoshka dolls. Confirmation would come from the investigation called “WiggleZ Dark Energy Survey“, led by a young Australian doctoral student, Morag Scrimgeour and his colleagues at the University of Western Australia in Perth. Using the Anglo-Australian Telescope, researchers have pinpointed the locations of 200.000 galaxies that fill a cubic volume 3 billion light-years on a side. It deals with the structure of the universe at scales larger than any previous investigation. Scientists have found that matter is extremely uniformly distributed throughout the universe, on extremely large scales, with little sign of fractal-like patterns. In some ways you could say that the universe is a lot like snow: composed of fractal flakes to be sure, but transitioning to a uniform sea of white as you move further away.
Once again let us focus on the contributions made by individual researchers at this level and on the global conclusions we have been able to arrive at. The participants in the scientific debate tenaciously defended their position, but always and only with the data and with the most appropriate reading of them. Nobody tried to cheat and nobody escaped the so-called "peer review“. The place chosen for the discussion was the most correct one, there was no attempt to create fans by speaking to non-experts and no one dared to epistemically transgress their own field of experience and study. This is generally how a scientific comparison. Any other type of comparison is not a useful debate, but a simple dialectical competition, a theater of words, where the catchiest or most elegant sophistry prevails and not the most solid argument. Contrary to what happens in an ordinary debate, in a healthy and constructive scientific debate it is not sufficient to state a hypothesis to have the right to speak, but it is required to do so in such a way that it can be weighed and evaluated with the scientific method. There is no principle of authority or democracy, only the method counts, the scientific one, that is, the possibility of arriving at a thesis - once a hypothesis has been formulated - through a logical-mathematical procedure and/or the analysis of experimental facts.
