What do Pirandello, math and a river have in common? - Part 2
With this series of three posts, I would like to explain to you a subtle common ground among three different areas of human knowledge: Pirandello (Italian literature), math, and nature (or better, natural science). In the first post (if you missed it, you can find it here), I have already unveiled this “mysterious Ariadne’s thread” that connects these three areas. In addition, we started talking about mathematical equations with complex numbers, butterfly effect, and fractals.
For introducing this second post, I would like to start again from fractals and from Mandelbrot (do you remember him?). Indeed, one of his most famous quotes (that became also the title of a documentary, on Netflix, about Mandelbrot) is the link between math and our next topic, which is science and nature. This sentence reads “why is geometry often described as "cold" and "dry"? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor is the trajectory of lightning a straight line.”
Chaos and fractals in nature
Chaos theory and fractal geometry have opened up previously unthinkable correspondences between the abstract operations of mathematics and the natural forms of our planet. Everywhere, on the Earth's crust or in broccoli, chaos and nature create and sculpt branched and fragmented landscapes and forms, in which details nestle within other details, within other and so on.
Broccoli’s fractal structure
Fractal in nature. The Gange’s delta.
Photo by: NASA Marshall Space Flight Center (Archive: Nasa, Space Shuttle, 11/19/05)
When chaotic phenomena shape the environment (sea backwash, atmospheric turbulence, erosion) fractal forms such as jagged coastlines, tree branches, and winding river courses remain as evidence. It is now interesting to understand the modalities with which particular figures are generated, ascribable to fractals, even in nature: what happens, for example, when the morphology of a landscape is modeled by the flow of water on the surface?
Water erosion, sediments and fractals
Water erosion has the ability to shape the landscape always, in the same way, generating shapes (like the one below) that have similar details at every scale, just as happens in fractals.
The erosive activity of water is manifested initially with rains, that exert on the land a beating action. The waters that do not penetrate into the subsoil begin to flow disorderly on it, first forming a uniform surface veil and then dividing into many rivulets, which flow along the slope and dig a large number of small furrows intertwined between them. This process is primarily responsible for removing significant amounts of sediment from the soil.
As these waters flow downstream (becoming streams, rivers, etc.), they increase their carrying capacity and begin to erode and transport increasingly bigger debris (silt, clay, and then sand and gravel). The river also continues its erosive process in the lowlands to the sea, where deposition of transported sediment begins. Right here, at the mouth of the river, you can find typical fractal landscapes caused by the "random" sedimentation of debris. The sediment deposits begin to obstruct the course of the river, which is forced to modify its course by dividing into smaller and smaller arms.
How is it possible, however, to trace these natural landscapes back to fractals?
The explanation can be found in mathematics: to transport the sediments, the current must perform a work that depends on the speed (V) and the flow rate of the watercourse (Q) according to the formula:
Ptot = Q x V2
since speed and water flow rate are two variable parameters, with this formula we can describe a multitude of cases and so this leads us back to those simple mathematical equations that the chaos theory wants to use to explain complex natural phenomena.
Conclusion
So, I am sure that now you are as amazed as me to discover that chaos, math, and nature are linked. You can also start searching for new fractal forms around you and looking at things from a different point of view.
The word chaos, however, has not only a value related to the world of science and mathematics. In fact, we can speak of chaos also referring to the discomfort, to the inner conflict of man experienced in the early decades of the twentieth century and caused by the great political and economic disruptions that have characterized this period. Of course, to keep the suspense going, this will be the topic of my last post.
