- Sahaana V
Dose of STEM: Time to Let Our Souls Spiral with (Not Just Frozen) Fractals All Around
Writer’s Note: This article requires a ground-level knowledge of concepts such as complex numbers and functions which are taught in high school. While this article is not math-intensive, if you are unaware of some of these concepts, you may find some parts of it confusing. You may, however, read this article for pleasure as this is a brief introduction to the vast concept of fractals.
What are Fractals?
Some people may remember the mention of the word ‘fractals’ in Idina Menzel’s song Let it Go in the movie Frozen. But what does the word mean?
When anyone looks at a fern, they see many small and similar-looking leaves branch off a larger one — individual parts of plants look exactly the same as the entire plant. In mathematics, this property is called self-similarity; the shapes that showcase this property are called fractals. Self-similarity is a crucial property of fractals, and this refers to the infinite nesting of a shape on all scales.
Nevertheless, this property is not enough to define a fractal. We need to look at dimensions as well. How many dimensions does a line have? It is known that a line has just one, a square has two, and a cube has three. But what about a fractal like the Sierpinski triangle below? Mathematicians have found that the dimension of a Sierpinski triangle is about 1.585, which is not an integer!
How is that possible? This is one of the many weird properties of fractals, and this dimension is called a fractional or Hausdorff dimension. Thus, another way to define fractals is 'shapes having a non-integral/ fractional/Hausdorff dimension’. Some famous fractals include, of course, the Sierpinski triangle, the Apollonian gaskets and the Mandelbrot set which would be discussed in the later sections of this article. Let us go back in time to understand these intriguing structures.
The Julia & Mandelbrot Sets
The term fractal was coined by a mathematician in the IBM research centre, Benoit Mandelbrot, in the year 1975, and he derived it from the Latin word fractus — which means broken or irregular. However, many mathematicians were working on fractals even before this term was introduced. In the year 1918, two mathematicians, Gaston Julia and Pierre Fatou, worked independently on iterations on a complex plane using the equation f(z) = z^2 + c, where c is a constant. For every value of ‘c’, this equation produces a fractal, i.e. when we plot every value of f(z) on a complex plane — similar to a Cartesian plane i.e. the ordinary x-y plane but for complex numbers (numbers of the form ‘a + bi’, where i = √-1) — we get fractals. The set of points is called the Julia set. Unfortunately, at that time, it was difficult to visualise what the Julia set of points would look like because the mathematicians did not have the technology required for that. This is where Mandelbrot, the father of fractal geometry, comes in.
In 1980, Mandelbrot was building upon the Julia sets with the technological capabilities of IBM computers and tried to iterate the equations, i.e., he used the equation’s first output as its next input. With the computers, he was able to crunch and manipulate the numbers a million times over and graph the output. What he got was a peculiar squashed bug-like image (see below) but with beautiful intricacies. This structure is now called the Mandelbrot Set which is made up of many values of ‘c’ for which the sequence starting at z = 0 does NOT go to infinity. This complex and beautiful structure is the most popular example of a fractal. Its Hausdorff dimension is 2 (Confused? Check this article by NewScientist on the Set's integral dimension.)
Koch Curves / Snowflakes
The Koch curves or snowflakes are also well-known fractals. The Koch snowflakes are generated by the sides of an equilateral triangle. A triangular protuberance is constructed at the central third of the side, and this process is iterated continuously, forming a snowflake-like fractal. This snowflake is based on the curve where iterations take place on a single line. The Hausdorff dimension of a Koch curve is 1.26 and is hence a fractal.
How do we get the Hausdorff dimension? In a Koch curve, at every iteration, four self-similar segments are each ⅓ the length of the original segment. Therefore, 3^δ = 4 and δ = log(4)/log(3) which is equal to 1.26.
Sierpinski Triangle and Apollonian Gaskets
A Sierpinski triangle (as seen above) is another fractal generated from an equilateral triangle (the starting object is often called the initiator; for example, the initiator of a Koch curve is a line) repeatedly divided into smaller triangles. It is made of three self-similar triangle pieces corresponding to the three functions that are iterated. The Polish mathematician Waclaw Sierpinski was the first person to think about the properties of this triangle, and therefore, this triangle is named in his honour.
Alternatively, Apollonian gaskets consist of circles tangent to each other. The first mathematician to consider this concept was Apollonius of Perga, a Greek mathematician and astronomer who lived in the 3rd century B.C. The Apollonian gaskets were the earliest known fractals and its Hausdorff dimension is approximately 1.30568. An Apollonian gasket exists for every value of 'n' such that the curvatures or the bends (the reciprocal of radii) are of the form −n, n + 1, n(n + 1), n(n + 1) + 1.
An Intersection Between Mathematics and Art
While fractals are prevalent in art, the fractal repetition of patterns is an artistic challenge. The famous Japanese artist Katsushika Hokusai’s painting, The Great Wave of Kanagawa,
(which was painted in the year 1820), is a great example of the use of fractals in art. In the painting, a massive ocean wave is painted as breaking off into smaller self-similar waves.
Another example is by an artist many people have definitely heard of — Leonardo Da Vinci. His drawing, Turbulence, is actually a diagram that he drew while studying fluid mechanics. This drawing clearly displays fractals in the ripples and bubbles of water.
Modern artist, Jackson Pollock, is famous for his fractal drip paintings. Many people have attempted to fake Pollock’s fractals and have failed.
Fractals are one of the most intriguing concepts that conjoin mathematics and art. They are widespread in nature as well — we can see them on Romanesco broccoli, pine cone seeds, rivers and coastlines (from an aerial view), branches of a tree, leaves, and of course, snowflakes.
Applications of Fractals
A lot of people think that fractals are pretty but have no practical use in real life. As a matter of fact, they do. For example, in the field of Computer Science, fractals are used extensively for image compression. Through fractals, images are compressed much more than by the usual ways, like the JPEG formats. The other advantage of fractal image compression is that when the images are enlarged, there is no pixelation! Fractals are used in the field of fluid mechanics to study turbulence in water (as Da Vinci did), as well as petroleum sciences to study porous media. In telecommunications, scientists have come up with the idea of a fractal-shaped antenna, to create a lightweight and smaller alternative of the conventional antennae. With the concepts of fractal geometry, much of the phenomena witnessed in nature can be visually modelled. For example, we can use fractals to model soil erosion and analyse seismic patterns too. Fractals showcase great complexity arising from simplicity.
“Fractal geometry is not just a chapter of mathematics, but one that helps every man to see the same world differently.” ~ Benoit Mandelbrot
Sources:
Fractals and the Fractal Dimension (Vanderbilt University): https://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html
M. Frame, B. Mandelbrot, N. Neger; September 2, 2021; "Fractal Geometry" (Yale University): https://users.math.yale.edu/public_html/People/frame/Fractals/IntroToFrac/InitGen/InitGenKoch.html
Fractal Geometry (IBM): https://www.ibm.com/ibm/history/ibm100/us/en/icons/fractal/
Krzysztofik, Wojciech. (2017). Fractals in Antennas and Metamaterials Applications. https://doi.org/10.5772/intechopen.68188.
Image Sources:
Da Vinci's Turbulences: https://commons.wikimedia.org/wiki/File:Etudes_turbulences_-_L%C3%A9onard_de_Vinci.jpg
The Great Wave of Kanagawa: https://www.publicdomainpictures.net/en/view-image.php?image=76219&picture=the-great-wave-kanagawa
Sierpinski Triangle: https://commons.wikimedia.org/wiki/File:Sierpinski-rgb.png
Koch Snowflake: https://commons.wikimedia.org/wiki/File:KochFlake.png
