Fractals and the practical implication of mathematics

Everything around us - the mountains, the clouds, the coastlines - are fractals. Benoit Mandelbrot, who died about a month ago at the age of 85, explains in an interview from 1983 what his discovery means

A year after Mandelbrot came out with his book The Fractal Geometry of Nature, fractals already excited mathematicians. In 1983 the mathematicians sat in front of the computer and peered at a strange black shape that appeared on the screen. The computer was not what it is today: a flat LCD screen, high definition and amazing color resolution, or a XNUMXD screen. Who ever thought of such a thing. It was a standard black-and-white screen, or rather, a green-and-white screen of the type familiar to mathematicians who worked in those years. The black spot, produced by the electron gun and appearing on the same screen, seemed to be an image of something, but they didn't know exactly what. It looks something incomplete and sickly, from which come out bristles, hills and all kinds of protrusions and additions of shapes. The appendages themselves looked like strange blooms. The form received the name "Mandelbrot group".

Benoit Mandelbrot spoke in 1983 with a journalist named Ed Regis. Below is the content of Mandelbrot's words from that 1983 conversation. He told Regis that he created the Mandelbrot Group, invented "the group", as he said and he added, "I have the honor to bear its name". In addition, and no less important of course, Mandelbrot began at that time in the new field of mathematics called fractals or fractal geometry. A fractal is an object whose shape is not smooth, such as a line, curve, or surface; but an irregular line, broken in any order of magnitude. Mandelbrot said, "I gave the name fractal from the Latin adjective fractus". Mandelbrot explained in those years that "the Latin verb franger means 'to break': to create irregular fragments". In 1983, some people at the Institute for Advanced Study in Princeton thought that fractals were the future of research.

Regis says that Mandelbrot expressed his manifesto in the eighties: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and shells are smooth, even lightning does not move in a straight line. In general, I claim that many forms in nature are so irregular and fragile that, compared to Euclid - the standard geometry - nature does not just present a higher level of complexity, but a completely different level of complexity."

Mandelbrot told Regis that while at the Institute for Advanced Study in Princeton as a young postdoctoral fellow in the XNUMXs, Mandelbrot wandered around the campus to see von Neumann's computer. Von Neumann designed and built a computer at the institute. But Mandelbrot didn't know how to use the machine yet, so he didn't bother with the computer. His computer business began later, when he discovered fractals while at IBM in New York.

Very early in his career Mandelbrot asked himself, "How long is the coastline of Great Britain?" His answer was that, in fact, there is no single answer that is correct. It all depends on the scale you choose to measure and also the measurement standard you choose. If we take a map and measure the distance between the northern and southern tip of Great Britain, we get a rough estimate of the length of the coastline. However, if we walk along the coast between the exact same two points, we will get a completely different answer. This is because we will cross the beach and go along every bump and small bay on the beach. If we were a tiny ant walking along the beach we would still be walking along an increasingly irregular path and therefore would measure a much greater distance between the two points north and south. And if we were from Microb, the distance would be even greater and so on...

At the time, Mandelbrot thought that this had implications for measuring the areas of countries, "the lengths of the borders between Spain and Portugal, or Belgium and the Netherlands, as reported in the encyclopedia of these neighbors, differ by 20 percent." Mandelbrot said in the XNUMXs, "We will not be surprised if a small country (Portugal) measures its borders more accurately than its larger neighbor." Luckily he didn't know the conflicts in the Middle East...

If we return to mathematics, the difference between Euclidean and fractal geometry is in the dimension. In classical geometry, the dimensions are expressed in whole numbers and a straight line has dimension 1, a plane surface has dimension 2, while a solid has dimension 3. But fractals are broken mountains and have broken edges and therefore have dimensions that are fractions, such as 1.67, 2.60, etc. Mandelbrot created such pathologies while working at IBM. “I came up with a suitable equation,” Mandelbrot told Regis, “and in 1973 we put together a very bulky plotter to create artificial coastlines. … Sometimes we had to sit up all night with the plotters. But when the first coastline finally appeared, we were all amazed. It looks just like New Zealand! Here was an elongated island, so square, and further from one side, two spikes that resembled Bounty Island... Seeing them had an electrifying effect on everyone... Now, after seeing the photos of the coastline, everyone agreed with me that fractals were part of the substance of nature."

Mathematicians in 1983 began working on the Mandelbrot group and enlarging the shape, and saw that one of its arms has another arm connected to it, which is exactly like it. It branches out from it in a zig-zag, like some kind of lightning strike. The whole image can even be enlarged and then you can even reduce it and get even smaller fibers, and they break at certain points. And if there is a winding line and you continue to zoom in, you find that there are more and more fibers coming out of this line and wandering in and out. You have to make a bit of an effort to see them, because they are on the edge of the ability to see, and in those days the computer could not enlarge more than that.

In the XNUMXs mathematicians tried to understand the structure and behavior of this thing, something they actually thought did not exist in our world at all. Realize that this is not another entity created by the computer that you see from time to time in hi-tech magazines, but something real, but at the same time something that does not belong to this world. The fact remains that the shape that was there on the computer screen is a mathematical object, a complete abstraction.

At that time it was thought that the fractal was the most complicated object in mathematics. Until then, the mathematicians were dealing with exact or clean entities, something that has logic behind it, a mathematical order, that is, the old and familiar mathematical order of classical mathematics and Euclidean geometry. But then came the fractal. He looks beautiful to the eyes. Mathematicians began to study it not because it is useful to someone, but because it is beautiful to the eye. "My motivation is first of all aesthetic," said Prof. John Milnor of the Institute for Advanced Study in Princeton in 1983 to Regis, when he studied the fractal, "I look at these things because they are beautiful in themselves." Milnor said at the time, “For some people the main motivation is that the research of these things might be effective. For me personally, benefit is just a happy byproduct." Milnor was nevertheless aware that the study of fractals had applications. At that time, the study of chaos burst into consciousness and with it the application of fractals to dynamic systems.

Milnor marveled that there was "still a level of possible magnification that is more subtle than what we're looking at now." While looking through the computer screen in the XNUMXs, Milnor said, "There is actually no inherent stopping point, and that is part of the mystery of the shape in front of us: the more you magnify each part of it, the more structures you will discover from within." It's like a never-ending series of Chinese boxes." This is the property of the fractal that it is self-similar. In other words, as if putting a mirror within a mirror on each smaller part of the fractal. Mandelbrot called this feature the "scale factor" (the reduction or enlargement of the shape, "scaling"), which means, the shape is the same shape no matter what level you look at the object.

In fact, mathematically speaking, looking at a fractal is a visual representation of a numerical function that is iterated over and over again. Mandelbrot explained to Regis, that "the Mandelbrot group is a group of complex numbers that have the characteristic, which perform a certain operation and then square. Take a number z, square z and add c. Then, square the result, and add c. Take the result and square it and add c, and do this many times. Basically, every time you get a result, you check to see if you left a circle with a radius of 2, and you plot it on a graph. As you continue, the group is drawn in more and more detail. But all you're actually doing is multiplying something by itself, and adding to itself: z squared plus c; all squared plus c; Everything squared plus c.”

Mandelbrot explained that the Mandelbrot group is the result of labeling the simple function (z2 + c) repeatedly when complex numbers are used as initial values. Mandelbrot said that "the Mandelbrot group especially exhibits extreme behavior". And he added, "The phenomenon that is characteristic of fractals is that there is a very simple formula and the result is an unusual complication. It's amazing to discover that one line of an algorithm, which doesn't look particularly interesting in itself, will lead to something with such an unusual structure."

At that time, fractals began to be thought of in the context of dynamic systems, turbulence and biological systems. They thought that perhaps the fractals could provide a model for the human lung and blood vessel system and Mandelbrot began to see fractals everywhere in nature - in all natural phenomena from the micro level to the macro level and in general to infinity... He began to produce equations on his computer that mimic the structure of natural phenomena such as trees, rivers , the human arterial system, the clouds in the sky, and various strange phenomena. Mandelbrot said that "fractals are the very substance of our flesh!"

In 1983 they started debating whether the fractal geometry is violated at the molecular level? Mandelbrot was enthusiastic and saw fractals everywhere, but still there is a limit. Milnor thought at the time, "If you are on a scale of up to ten molecules and more, the fractal geometry may be valid from here on up." However below this limit it was thought that the fractal geometry might be violated. The question asked was, "If when you inflate part of the Mandelbrot group repeatedly, do you reach the limit?" Mandelbrot was busy with his favorite creatures and called one of them "the dragon San Marco". He explained, "This is a wild extrapolation by the mathematician of the skyline of the basilica of Venice, together with the reflection in the flooded piazza." The dragon's algorithm is so simple, just twenty lines and run it. The cathode rays of the XNUMXs did the job and created a Venice cathedral that is reflected in the water. And since then the fractals penetrate the scientific studies everywhere...

Mandelbrot explains about fractals

Mandelbrot's 1982 book

The Mandelballev - three-dimensional Mandelbrot group