On October 14, 2010 the French mathematician died Benoit Mandelbrot, which was the visible head of the so-called theory of fractals. Mandelbrot developed a multidisciplinary activity recognized in several scientific fields, but a large part of its public reputation is due to the fact that one of the fractal sets (term coined) most represented bears his name.
There is no universally accepted definition of fractal, but virtually all authors link this term to some form of self-similarity and to fractional dimensions. Self-similarity is the property that guarantees that the same structure is conserved at different scales. An example that is easy to find in the market is the romanesco broccoli. Each small portion of this vegetable reproduces its global form. Simpler examples are the branches of a tree or the blood vessels, in which the smallest divisions are a scale model of the larger ones. On a more abstract plane, there is Sierpinski's triangle. If in an equilateral triangle we mark half of each side, when joining marks on opposite sides, the triangle will be divided into four. Discard the central one, the one that is inverted, and repeat the process indefinitely with the three remaining triangles. The result is a set that observed with a magnifying glass of any magnification has the same appearance.
The dimension is a more elusive concept. Normally we consider dimension as the number of parameters we need to describe something. So we say that space is three-dimensional because we need three coordinates, long, wide and high to indicate each point. This notion of dimension does not describe well the case of rough objects, in which very abrupt changes occur. In this case, the dimension can stop being a whole number and become a fractional value.
Mandelbrot studied this situation in one of his most famous articles: How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension (affordable for anyone with baccalaureate knowledge). When measuring the length of a very irregular coast, the result will depend on the size of the measuring rule used. If a very small one is used, all the nooks and crannies will cause the length to increase considerably. That is very different from what would happen for a smooth curve like a circumference (in which we will always obtain in the limit the usual 2 pi R). The coast of Great Britain, affirmed Mandelbrot, came modeled by a geometric object called fractal. It is a curve, in the sense that it can be described with a continuous function, but the dimension of its image is not one-dimensional, in some sense, as it is in a smooth curve.
The fractal dimension is determined by the variation of the measured length, in terms of the variation of the rule. It is said that a curve has dimension D if the values obtained by multiplying the length measured by a power D-1 of the length of the rule used are approaching a constant for small rules. For dimension D = 1 there is independence from the rule, provided it is small, while for dimension D = 3/2, every time we reduce the rule to a quarter, the length measured will be multiplied by two. In his article, Mandelbrot affirms that the fractal dimension of the coast of Great Britain is 1.25, a value "more fractal" than, for example, that of the border between Spain and Portugal (which is 1.14 according to data of the mathematician English Lewis Fry Richardson).
Beyond its applicability in areas such as medicine or communications, fractals have had a remarkable popular success, which is that they have an immediate visual beauty based on symmetries, such as the work of M.C. Escher. This popularization of fractal images is undoubtedly linked to the development of information technology. Nowadays it is so easy to generate them in any computer (repeating millions of times the process that gives rise to self-similarity), that it is not surprising that the images have traveled from the university texts of mathematics to the folders of the adolescents through infinity of supports digital
Fernando Chamizo He is a tenured professor at the Autonomous University of Madrid and a member of ICMAT. Agate Rudder is responsible for Communication and Dissemination of ICMAT
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions, and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".
Editing and coordination: Agate Rudder (ICMAT)