Alberobello (Iarubbèdde in local dialect) is an Italian town of 10,237 inhabitants in the metropolitan city of Bari in Apulia (Southern Italy), part of the Itria Valley 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 December 6, 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 concrete materials.
Among all these “dry” works reign the typical low walls: real physical boundaries of estates, roads, state-owned spaces.
A saying, we shall shortly see that cannot be disproven, goes: you will not find one dry stone wall similar to another; they are like fingerprints because of the unique interlocking of the limestone rocks of which they are made.
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But, they can be reproduced as a fractal model
A researcher at the University of Bari ( Department of Mathematics) , Giuseppe G. Monno, in 1986 posed a simple problem, despite not having perfectly identical structures between dry stone walls: could we derive a universal model for eventual reconstruction or construction?
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What are Fractals and what are they used for?
What are fractals? Traditional mathematics loved simple shapes: straight lines, squares, regular polygons the pentagon, hexagon, octagon; circumferences and parabolas, spheres, cubes, circles, ellipses and so on.
In nature, on the other hand, regular figures are pure exceptions: where in nature is there a cube, or a perfect sphere? Nature has always enjoyed playing with man. Trees, clouds, ferns, cauliflowers, lightning and thunderbolts, mountains and rocks, the coastlines of countries and nations: everything appears irregular, angular … fractal. In the attached figures I proposed some photos of natural fractals. From a formal point of view, there is no all-encompassing definition of a fractal.
A fractal figure (from Latin fractus) is a jagged, broken, angular figure. Benoit Mandelbrot gives a number of definitions: – in 1978: a fragmented, broken, highly discontinuous shape or figure; – in 1982: a whole for which the dimension according to Hausdorff and Besicovitch strictly exceeds the topological dimension; – in 1986: a shape made up 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 heights and variations on the 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 Norway's coastline with all the inlets, peninsulas, islands and islets and fjords! The calculation is yes 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) from very simple rules of construction; likewise, fractal geometry can provide the length of the coast of Norway, as long as you tell a priori what the length of the ruler with which you want to measure it is
What are fractals for? What is mathematics for? Geometry? What are derivatives for? Integrals? Differential calculus? They are for calculus, for describing nature and phenomena: in short, for describing and simulating 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 borders of states according to their degree of “fracture,” of “edge”: from the coasts of South Africa, to the borders of Germany, to the coasts of England to those of Norway. It must be said that fractals would not have become so imperious if there were no large computers; in fact, very large memories are needed to be able to do 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 DRYWALL BY MEANS OF FRACTALS
There is no strict definition of a fractal, so it is said that “fractals are objects with an irregular, jagged shape” . Then one can distinguish whether a given object is a fractal or not by analyzing their characteristic properties, which are “self-similarity” and “fractal dimension” itself. Self-similarity or self-similarity is that property that some objects, particularly natural objects such as clouds, mountains or stones, have, in which not only can details be obtained at (theoretically) any magnification factor, but these details are also such that the degrees of irregularity corresponding to different scales are supposedly equal.
One is then led to believe that, minus the scale, the same mechanism has generated both the minute details and the global characters of the object. In the case where each piece of the whole is ( statistically speaking) equal to the whole, one will say that the object is self-similar.
The problem found to be solved on several occasions was that of analyzing a wall and visually simulating its shape.
Regarding the first point, the research was aimed at finding a characteristic parameter not so much of the single wall as of a certain set of walls. To this end, attention was focused on calculating the fractal dimension of the walls considered , keeping in mind the result of previous experiments, from which it was 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 fractal dimension can prove effective, for example, in the classification and creation of archives, as it offers a discriminating criterion, although it is necessary to take into account the fact that, since, the dimension calculation software considers the ratios between stones and the gaps between them, and does not take into any account the particular geometry of the stones themselves, a wall composed of irregular stones and one composed of regular bricks may give the same value of fractal dimension, ie, the numerical amount of dimension does not contain all the information about the wall under consideration.
Solutions … fractals ( alert: nerdy stuff )
The fractal technique offers an ideal solution. In fact, because the resulting ensembles are fractals, and as such have the property of being self-similar at all scales, they offer what a simple memorized image can never give, namely, the possibility of enlargements at will on the ensemble with ever- new details.
One technique takes advantage of the Collage Theorem and allows the reconstruction of an object very close to the starting set by condidering a system of iterated functions w1, w2 … wn where each wj is an affine transfromation that consists of the starting set appropriately rotated and scaled and put in such a way that the set of all maps completely covers the set itself, ie, 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 in the fact that it is the task that falls to the operator who must optimally identify the affine maps.
To overcome this incovenience, another technique was sought, which is the one implemented, exploiting 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 computable and reproducible fractal dimension.
First the software draws a Brownian motion of given fractal dimension, which reproduces the upper contour of the wall. Then “break” points are randomly chosen from which, with appropriately chosen angles, depart (again with Brownian motions of the same initial type) what become the sides of the stones of the first level.
Then given the broken line formed by the lower contours of the stones of the first level, which still forms a Brownian motion with the said characteristics, the reasoning is repeated, obtaining a second level, and so on up to the number of levels chosen. The program given the aleatory character of Brownian motion draws a wall resembling the starting one, but with the same fractal dimension, thus with the same characteristics of interruption and irregularity.
Conclusion of AILoveTourism
Without going into hostile technicalities what is the conclusion of the work?
Two different fractal dimensions can be defined ( depending on whether one wants to privilege the geometry of the interstices or the ashlars). Thus, the use of fractals to reconstruct a reconstruction model of any dry stone wall is, fully, plausible.
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