Alberobello (Iarubbèdde in the local dialect) is an Italian town of 10 237 inhabitants in the metropolitan city of Bari in Puglia, part of the Valle d'Itria and the Murgia dei Trulli.
It is famous for its characteristic trulli, a constructive model of spontaneous architecture, declared a UNESCO World Heritage Site on 6 December 1996 ( ft. Wiki).
Inserted in the so-called Murgia dei Trulli, its landscape is surrounded, everywhere, by different buildings and architectural styles built with "dry" techniques; that is, without or with moderate use of cementitious materials.
Among all these "dry" works, the typical low walls reign: real physical boundaries of farms, roads, state-owned spaces.
A saying, which we will see shortly that cannot be denied, says: you will not find a dry stone wall similar to another; they are like fingerprints due to the unique interlocking of the limestone rocks they are made of.
science
But, they can be reproduced as a fractal pattern
A researcher from the University of Bari (mathematics department), Giuseppe G. Monno, in 1986 posed a simple problem, despite not having perfectly identical structures among the dry stone walls: could we obtain a universal model for a possible reconstruction or construction?
Spoiler: the answer came from FRACTALS.
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What are Fractals and what are they for?
What are fractals? Traditional mathematics loved simple shapes: straight lines, squares, regular polygons, pentagons, hexagons, octagons; circles and parabolas, spheres, cubes, circles, ellipses and so on.
In nature, on the other hand, regular figures are pure exceptions: where in nature is a cube or a perfect sphere found? Nature has always enjoyed playing with man. The trees, the clouds, the ferns, the cauliflowers, the lightning and arrows, the mountains and the rocks, the coasts of the countries and nations: everything appears irregular, angular ... fractal. In the attached figures I propose some photos of natural fractals. From a formal point of view there is no overarching definition of a fractal.
A fractal figure (from the Latin fractus) is a jagged, broken, angular figure. Benoit Mandelbrot he gives a series of definitions: – in 1978: a fragmented, broken, strongly discontinuous shape or figure; – in 1982: a set for which the dimension according to Hausdorff and Besicovitch strictly exceeds the topological dimension; – in 1986: a form made of parts that are somehow similar to the whole (1986). None is able to completely exhaust all possible cases.
Mathematics and geometry before Mandelbrot was unable to describe complex objects. For example, it is easy to calculate a volume: basis for pitches and variations on a theme; but try to calculate the "exact" volume of a tree, or a bush or a sponge, or the bronchial system of a lung. Try to measure the length of the coasts of Norway with all the inlets, peninsulas, islands and islets and fjords! The calculation is easy but complicated, complex. Fractal geometry shows that, at least often if not always, it is possible to construct complex objects (which Mandelbrot calls sets) starting from very simple construction rules; likewise, fractal geometry is able to provide the length of the coast of Norway, as long as you say a priori what is the length of the ruler with which you want to measure it
What are fractals for? What is math for? The geometry? What are derivatives for? Whole grains? Differential calculus? They are used to calculate, to describe nature and phenomena: in short, to describe and simulate nature.
Fractal geometry expands the power of classical geometry invented by Euclid around 300 BC to non-integer dimensions, introducing a huge variety of applications.
Fractal geometry knows how to classify the coasts and the borders of the States according to their degree of "fracture", of "gleaning": from the coasts of South Africa, to the borders of Germany, to the coasts of England up to those of Norway. It must be said that fractals would not have imposed themselves in such an imperious way if there were not the great calculators; in fact, very large memories are needed to be able to make the most sophisticated and interesting simulations.
Paesini di Natale tra Puglia e Basilicata
i paesini di natale 3 x 1 Tre Audioguide per smartphone al prezzo di una sola.Anzi,…
Are dry stone walls eco-friendly? Yes, the Terracing Borders of Life and Death ( Part 2 )
Are drystone walls eco-friendly? Yes, right from birth. (Part 1)
Are dry stone walls eco-friendly? Yes, since birth. (Part 1)
Adventure Tourism in Apulia: “the blades” of the territory.
Adventure Tourism in Puglia: the "blades" of the Territory
GRAPHIC RECONSTRUCTION OF A DRY WALL BY MEANS OF FRACTALS
There is no strict definition of a fractal, so it is said that “fractals are irregularly shaped, jagged objects”. Then one can distinguish whether or not a given object is a fractal by analyzing their characteristic properties, which are “self-similarity” and the same “fractal dimension”. Self-similarity or self-similarity is that property that some objects have, in particular natural objects such as clouds, mountains or stones, in which not only can particulars be obtained at (theoretically) any magnification factor, but these particulars are also such that the degrees of irregularity corresponding to the different scales are more or less the same.
One is then led to believe that, apart from the scale, the same mechanism has generated both the minute details and the global characteristics of the object. In the event that each piece of the whole is (statistically speaking) equal to the whole, the object is said to be self-similar.
The problem that we found ourselves solving on various occasions was that of analyzing a wall and visually simulating its shape.
As regards the first point, the research was aimed at identifying a characteristic parameter not so much of the single wall, but of a certain set of walls. For this purpose, the attention has been focused on the calculation of the fractal dimension of the considered walls, keeping in mind the result of previous experiments, from which it has been deduced that the equality of the dimension in the various cases suggests that, despite their different aspects, there is a common source for the data.
The calculation of the fractal dimension can be effective for example in the classification and in the creation of archives, as it offers a discriminating criterion, even if it is necessary to take into account the fact that, since, the calculation software of the dimension considers the relationships between the stones and the interstices between them, and does not take into account the particular geometry of the stones themselves, a wall made up of irregular stones and one made up of regular bricks can give the same value of the fractal dimension, i.e. the numerical quantity of the dimension does not contain all the information relating to the wall in question.
Solutions … fractals ( alert: nerd stuff )
The fractal technique offers an ideal solution. In fact, since the sets obtained are fractals, and as such have the property of being self-similar at all scales, they offer what a simple memorized image will never be able to give, namely the possibility of zooming in at will on the set with particular always new.
A technique exploits the Collage Theorem and allows the reconstruction of an object very close to the starting one, considering a system of iterated functions w1, w2 … wn where each wj is an affine transformation consisting of the starting set suitably rotated and scaled and put in such a way that the set of all the maps completely covers the set itself, i.e. so that the Hausdorff distance between the figure obtained from the set of maps and the starting set is as small as desired. The problem with this method is that it is the operator's task to identify the affine maps in an optimal way.
To overcome this inconvenience, another technique was sought, which is the one implemented, which exploits the idea that the network of lines obtained by considering only the interstices between stone and stone, is very close to a Brownian motion of calculable fractal dimension and reproducible.
First the software draws a Brownian motion of given fractal dimension, which reproduces the upper outline of the wall. Then some "interruption" points are randomly chosen from which, with suitably chosen angles, those that become the sides of the stones of the first level depart (always with Brownian motions of the same initial type).
Then, having considered the broken line formed by the lower contours of the stones of the first level, which still forms a Brownian motion with the aforementioned characteristics, the reasoning is repeated, obtaining a second level, and so on up to the number of levels chosen. Given the aeatory character of Brownian motion, the program draws a wall similar to the starting one, but with the same fractal dimension, therefore with the same characters of interruption and irregularity.
Conclusion of AILoveTourism
Without going into difficult technicalities, what is the conclusion of the work?
Two different fractal dimensions can be defined (depending on whether you want to favor the geometry of the interstices or of the ashlars). Therefore, the use of fractals to reconstruct a reconstruction model of any dry stone wall is, fully, plausible.
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Alberobello, Bari. ( Italy )
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