Fractals - A Map to the Universe
Fractals are stunningly beautiful.
Their perfect, hypnotic swirls are so alluring that they have found their way into popular culture, decorating textiles, T-shirts, books, calendars, mugs, greeting cards, posters, screen savers and every other product imaginable.
And yet, for all their beauty, fractals are not creations of artistic whim, but pure mathematics.
Beyond their undeniable aesthetic appeal, fractals are complex mathematical equations that also have practical uses, both in science and in everyday life.
But fractals are not just constructions borne out of mathematics - they are equations to measure and describe what is already all around us. A fact not fully discovered but until the mid 1970's.
And it all began with Monsters...
Back in 1958, the budding computing company IBM had a technical problem that puzzled its brightest engineers.
During these early days of computing, the IBM engineers where trying to send data through telephone cables (a visionary technique that would later evolve into Fax machines and the Internet), but the data sent was lost at times and no one could figure out why, or how to solve this issue.
Polish mathematician Benoit Mandelbrot took all the data collected from these attempts, and graphed it. Astonishingly, he found that, regardless of the time lapse he picked (1 hour, 1 minute, 1 second), the graph image remained virtually the same. This stunning result reminded him of old mathematical paradoxes called “Monsters”.
These were mathematical equations of complex numbers that rendered odd, unexpected results.
Back in 1883, the German mathematician Georg Canter produced the first of these monsters, when he decided to divide a line into three equal parts, taking the middle part out and repeating the same process with each consecutive line.
He called this a “Canter Set”, and very much resembled the graphs obtained by Mandelbrot at IBM.
Although in visual terms Canter’s experiment would stop after a few snippets and subdivisions, in mathematical terms the experience is infinite - that is, one could go on forever dividing the first initial line into smaller and smaller bits.
In 1904, the Swedish mathematician Helge von Koch came up with another Monster in what became known as the “Koch Curve”or , more popularly, the “Koch Snowflake”.
Koch started with an equilateral triangle, on which he added two angles on each straight line, repeating the process over and over. Whenever there was a straight line, he’d add two more angles.
Once again, although visually we could stop the process after a few repetitions, mathematically the process can go on infinitely.
What’s even more intriguing about this construction is that, although it looks like a finite construction (a finished ‘snowflake’), and its area can be determined, it’s perimeter goes on infinitely.
However, these oddities were no more than mere curiosities without any practical use, until 1967, when Benoit Mandelbrot published a now famous paper with a seemingly simple question: How Long is the Coast of Britain?
Easy enough, one could think. Just take enough topographical measurements and you’ll have the answer, right?
The problem that Mandelbrot showcased, however, was that there didn’t seem to be a single answer to this question. Each measurement made in the past had rendered a different answer, depending on the method of measurement used.
What’s more - the more detailed the measurement method used (measuring every crook and cranny of the coastline as opposed to a broader approximation) rendered a much larger area result. The experts were flabbergasted. How could that be? Isn’t a coastline a defined physical space determined by a finite area? And, even if there were differences, the overall area should remain about the same.
All these mysteries already weighed on Mandelbrot’s mind on the face of the IBM problem, and he decided to focus on these mathematical paradoxes more in depth.
In 1975 he published the fruit of his studies: “Fractals, Form, Chance and Dimension”. In the volume, he renamed these mathematical paradoxes as ‘fractals’(from the Latin ‘Fractus’, fractured).
Fractals, explained Mandelbrot, were curves that were irregular all over, and that present the same degree of irregularity at any scale.
Mandelbrot also presented a new mathematical construction, based on an earlier construction by French mathematician Gaston Julia.
The construction proposed by Julia in 1918 - called a “Julia Set”- was, in very simple terms, a formula in which we enter a complex number, calculate a result, and re-enter this result into the formula. As in the previous “monsters”, the process is repeated over and over again.
Depending on the value of the number entered, the formula would render a different shape when graphed.
In 1978, aided by the new computing technology, Mandelbrot created a new equation combining all Julia Sets, and set out to calculate millions of its reiterations, graphing them into one single image.
The result was stunningly beautiful, complex and endless. It was called the “Mandelbrot Set”.
Just like its predecessors, the Mandelbrot set contains endless repetitions of itself in perfect detail, advancing into an infinite progression that begins where it started, and continues without end.
The world was bowled over.
Progression of zoom-ins into the Mandelbrot set
But not all scientists were convinced by Mandelbrot’s claims. Many saw no practical use for his findings. Fractals, they said, were “pretty, but pretty useless”.
Undeterred, in 1982 Mandelbrot published a second book: “The Fractal Geometry of Nature”. In it, he postulated with clear visual examples that fractals are not mere mathematical ‘oddities’, but they abound in nature: in rivers basins, clouds, trees, ferns, the venous system, mountain ranges, lightning bolts, coastlines, and many more.
In fact, if we take a tree as an example, and we zoom in closer and closer into more detail, we’ll notice that the tree’s original ‘design’ is repeated over and over, from its full size to the branches and roots, and all the way down to the tiniest leaf veins.
The same happens with a broccoli: if we take a head of broccoli and cut it in half, and then proceed to cut it into smaller and smaller pieces, we’ll see that even the tiniest resembles the initial ‘design’ of the whole broccoli.
This is also the case with clouds, sponges, mountain ranges, coral reefs, river deltas, the lung structure, cobwebs, etc. All these contain within themselves copies of their same ‘design’ repeated over and over.
This endless copying (called “reiteration”) is at the heart of fractal geometry.
It was not that fractals were ‘wrong’, Mandelbrot explained, but that the geometry used until then to describe the natural world, was not the right one for it.
This new perspective forced mathematicians all over the world to rethink the use of the traditional Euclidean geometry applied to the natural world.
Fractals are simple, economic and efficient, three attributes favoured by nature, which explains its abundance. But, is there any practical use to this ‘Fractal Magic’?
Of Gaskets and Carpets
In 1915, Polish mathematician Waclaw Sierpinski created a simple fractal starting with an equilateral triangle, from which he took out an equilateral triangle from its midst. He would then take out an equilateral triangle from the resulting triangles, and would repeat the process over and over again.
Another similar construction is the Sierpinski Carpet, in which the triangle is replaced by a square, following the same procedure.
First 5 iterations of the Sierpinski Carpet
He also created the Sierpinski Curve, a defined sequence of continuous fractal curves.
Now, take a look at these telecommunication antennas - notice anything familiar?
In 1990, radio astronomer Nathan Cohen attended a lecture on Fractals by Mandelbrot in Hungary. An avid ham radio listener, Cohen had a problem: his landlord didn’t allow him to rig an antenna from his apartment. But during Mandelbrot’s lecture, an idea sparked in his mind: what if he applied fractal theory to a radio antenna?
He experimented with a hand-made test version, similar to Koch’s curve, and the result was incredible. It had the reception of a much bigger antenna at a fraction of its size. He had created the first fractal antenna in history, and his radio problem was solved.
However, with the increasing popularization of cell phones, blue tooths, wi-fi and wireless telecommunications, Cohen’s problem became the world’s problem: how to guarantee clear reception in all portable devices without using a different antenna for each radio frequency?
The answer was: fractals.
In fact, math calculations have confirmed that fractals are not merely a way to solve this problem, but the only way to solve it.
Nowadays, fractal antennas are even being developed in three-dimensions in order to amplify their reach, such as the Menger Sponge, a construction conceived by the German mathematician Karl Menger in 1926, based on the Sierpinski carpet.
Interestingly, fractal patterns had already appeared in ancient designs all over the world, often thousands of years before the math was developed to produce them through equations.
Fractals are related to the Golden Section and the Fibonacci numbers in that all these present mathematical progressions already found in nature, but the practical application of fractals is much wider.
Aside from their use in telecommunications, fractal science is being applied to diverse areas, such as physics, geology, meteorology, astronomy, engineering, computer science, architecture, the visual arts, music, economics, the stock market, etc. In fact, we have only just begun to discover the potential uses of fractal science in our lives.
For instance, engineers have designed fractal coolers for fragile liquids, and fractal mixers for liquids that require extreme precision mixing (such as in the oil industry). Fractals are also being studied in architecture to increase the efficiency in the design and planning of cities.
In medicine, fractal systems work alongside medical image programs (such as MRIs) to differentiate with precision healthy from sick cells in dense and complex tissue, something of vital importance in the detection and treatment of diseases such as cancer.
Other medical uses include genetics, anatomy, morphology, etc. to quantify, describe, diagnose and even help conditions such as arrhythmia, diabetic retinopathy, problems of the neural systems, and more.
Fractals are used widely in computer graphics ever since in 1978 Loren Carpenter created the first computerized mountain range for Boeing enterprises, followed in 1982 by the first GCI sequence for a feature film in “Stark Trek II: the Revenge of Khan”.
From hair textures to detailed landscapes, basically any construction that requires endless layers of repetition -such as the lava bursts in “Star Wars: The Revenge of the Sith”- applies fractal calculations.
Fractal geometry is used widely in computer science, and fractal image compression is one of the most popular tools, since images retain the same level of detail at all sizes.
Also, since fractals are used to describe the roughness of a surface, they are extremely helpful in geology, topology and the study and development of new materials.
In astronomy, a novel idea -still in development- that the entire Universe may be fractal, would force astrophysicists to reconsider all their theories to date, including the Big Bang.
Fractals may even provide a key to one of the most puzzling mysteries of the universe: the chaos theory. The study of chaos and fractals is a new field that unites mathematics, theoretical physics, art and computer science, and may lead to a new science altogether.
Video - Fun With Fractals (5:03)
https://youtu.be/XwWyTts06tU
It’s a common concept that science and art are mutually exclusive, but Fractals combine the best of both worlds.
Perhaps due to their intrinsic visual nature, fractals reveal to all of us what mathematicians have been claiming for centuries, as if it were a ghost that only they could see : Maths are beautiful. And math formulas are only the numerical expression of the wonderful universe around us.
Did you know that...?
Did you know that there are fractals in three dimensions? Mandelbulbs are mathematical constructions, manifestations of the mandelbrot equation in three dimensions.
Although there is no three-dimensional analogue of the two-dimensional space of complex numbers (which are required to construct a fractal), it is possible to build a mandelbrot set in four dimensions using bicomplex numbers and quaternions (both extensions of complex numbers) plugged in different formulas.
The first mandelbulb was created in 1997 by Jules Ruis, and this was later developed by Daniel White and Paul Nylander. In 2010, Tom Lowe added another construction with the Mandelbox, and as the capabilities of our software evolves, so do the range of fractal objects we can create.
Just as their two-dimensional originators, Mandelbulbs are fascinating to watch, as in this video : https://www.youtube.com/shorts/f8ek83Z9SFo
To Learn More...
* Can't have enough of fractals? Check out this website, where you can become a member to keep abreast of the latest in 3D fractal designs, and even find tutorials to create your own fractals!: https://www.mandelbulb.com/
Related Articles
If you enjoyed this article, consider Sharing it, or Subscribe to receive similar articles once a month in your inbox.
Sources: Book: “Complexification: Explaining a Paradoxical World Through the Science of Surprise” by John L. Casti (Harper Perennial); NOVA Documentary: Fractals, The Hidden Dimension; Fractal Foundation.org, Encyclopediaofmath.org, Mathigon.org, Chemnitz University, Wikipedia.
Comments
Post a Comment