The Fractal Geometry of Nature

The history of fractals is not long. It began in 1975 with the revolutionary work of mathematician Benoit Mandelbrot, “A Theory of Fractal Series”, which later became his manifesto book “The Fractal Geometry of Nature”.

Mandelbrot invented the word fractal to reunite the work of many others before him. Mathematicians such as Waclaw Sierpinsky, David Hilbert, George Cantor şi Helge von Koch created the first fractals, generally as abstract exercises, without having any idea about their significance. They identified them as bizarre forms which entered into a contradictions, a rather unpleasant one, with their concepts of space, area, distance and size. To unify the forms under one name Mandelbrot invented the word fractal, from the Latin word “frangere” which means “to break into irregular pieces”. Mandelbrot also noted: “because the word algebra derives from the Arabic jabara (to bind together), between the words fractal and algebra there is an etymological contradiction”.

After working a while with the fractals, Mandelbrot discovered that similar iterative (repetitive) processes can produce abstract mathematical constructions known as the Mandelbrot set and the Julia set. Like other fractals, these sets have been discovered long before Mandelbrot, but they were so complex that they required powerful computers to observe and study them.

The fractal object can be opened in similar parts to the original object

And yet, what is a fractal? The simplest definition is probably this: repeating to infinity and on multiple hierarchical layers of geometrical patterns, which are based on a mathematical relation from the category of the “golden ratio” (Pi number 3.14, The Fibonacci number, the basis of the natural logarithm). The fractal object can be opened in parts that each are similar to the original object (self-similarity). A fractal has an infinity of details that are presenting in a sequence that is repeating.

From Mandelbrot’s fractals to Fournier’s ice crystals, to Sierpinski’s triangles, the blood vessels, tree leaves, butterfly wings, fern leaves, snail shells, to the cauliflower, sunflower, the sand dunes of the desert, the DNA, corals, coastal ranges, heart rhythms, the water, wind and even music… it is actually a repetition of motifs, after a certain algorithm and certain rules governing the Universe.

The fractal geometry has entered not only in physics but also in medicine, climatology, geology, seismics, cinematography and even marketing and economics, all using fractal simulation programs. Did you know that there is fractal music, or fractal art?

With the help of Mandelbrot’s simulations it was possible to predict the variation in the price of cotton exchange, which suddenly made fractals more interesting.

In medicine there are applications in brain activity modeling, kidney, lung structure etc.

Everything is perfect as it is

Psychologists speak of so-called dynamic diseases that occur when fractals are desyncronizing, or when a person is no longer in harmony with the Universe.

Geneticists are convinced that the DNA molecule is one of the most complicated fractal pattern existing in nature and is par excellence that “self similarity” as well as the “part like the whole” principle.

Could it be the same reason why Eminescu said in the “Letter I”: “One is in all, as well as one is all”?

Nobody knows for sure how the spirals from the Julia and Mandelbrot sets are occuring from the simple non-linear ecuations and why they are following so closely nature’s designs. These themes are in the interest of current scientific and mathematical research. When a series of equations are left to their own winding iterations, mathematics itself seems to delight in the natural visual beauty. Could this be saying anything about God?

It has been noticed that people have a clear aesthetic preference for fractal images, regardless if those images are generated by nature, math or are manmade.

In his book, Mandelbrot noted that only Euclidean geometry can speak of complexity. The geometry of fractals is based on very simple sets. Does God think simply? If so, then why do we complicate ourselves so much?

Disorder is a part of people’s lives, since always. Mankind had to learn to measure disorder and fractals are the ones that reveal clearly, farely, an order in this disorder. These also showed that between order an chaos is a very fine difference. Is it so between love and hate?

The beauty within us extends beyond our comprehension, and when we will be able to bypass certain barriers of thought and egoism, we’ll notice with amazement that we have no reason to be unhappy! Because everything is perfect as it is.

Dr. Andreea Hefco

Source: Geometria Naturii.