Leonhard Euler
Matryoshka
Ece Paksoy
Let’s choose a number - for example 7. Then square that number - which gives us 49 in our case. Then square the resulting number and continue until we find a repeating point or a boundary. Unfortunately, if you had chosen a number bigger than 1, you would have seen how quickly it started to increase without any limits. However, if you have chosen a number between 0 and 1, like , the number becomes smaller and smaller and remains in a bounded place. But if we had chosen a negative number like -5, it would have also blown up and would have gone to infinity. So, which numbers do stay in place and remain in a bounded area when we constantly square the ending result?
The answer is given by the Mandelbrot set. The Mandelbrot set is the set of values of c in the complex plane for which
, where and
remains bounded. Thus, a complex number c is a member of the Mandelbrot set if, when starting with and applying the iteration repeatedly, remains bounded for all . Then,
is the set of numbers we get by plugging the numbers. If this series doesn’t diverge, then c belongs to the Mandelbrot set. If it diverges, then it does not belong to the set.
For example, for c = 1, the sequence is 1, 2, 5, 26, ..., which tends to infinity, so 1 is not an element of the Mandelbrot set. On the other hand, when c = −1, the sequence is −1, 0, −1, 0, ..., which is bounded, so −1 does belong to the set. Since we look at the Mandelbrot set in a complex plane, let’s check what happens when c is equal to the square root of -1, also known as i. Then the sequence is i, (−1 + i), −i, (−1 + i), −i, … which does not go to infinity, and, so it belongs to the Mandelbrot set.
If the series diverges (escapes to infinity) for a particular point , it is colored white (or color or intensity related to how fast the point escaped). If the point doesn't escape, the point is shaded black and is said to be inside the Mandelbrot set. When graphed to show the entire set, the resultant image is striking and pretty.
When additional colorings are added, we get beautiful pictures. The Mandelbrot set has become popular outside mathematics also for its aesthetic appeal. It is one of the best-known examples of visualization and mathematical beauty and motif.
References:
“The Mandelbrot Set.” The University of Utah, 1998, www.math.utah.edu/~alfeld/math/mandelbrot/mandelbrot.html.
McMullen, Faber. “The Boundary of The Period 1 Cardioid.” The Fractal Zone, sites.google.com/site/fabstellar84/fractals/cardioid.
Kamperis, Stathis. “An Almost One-Liner to Construct the Mandelbrot Set with Mathematica.” A Blog on Science, 1 Dec. 2020, ekamperi.github.io/math/2020/12/01/mandelbrot-set-one-liner.html.
Figure References:
/u/jeromeawhite. “Mandelbrot Set.” GeoGebra, 26 Apr. 2019, www.geogebra.org/m/mfewjrek.
“Mandelbrot Set Applet.” Princeton University, The Trustees of Princeton University, www.cs.princeton.edu/~wayne/mandel/mandel.html.
“Further Mathematics A Level.” Worcester Sixth Form College, 1 Oct. 2020, www.wsfc.ac.uk/courses/a-levels/further-mathematics-a-level/.
“Wild and Crazy Mandelbrot Set Full of Energy by Matthias Hauser.” Pixels, pixels.com/featured/wild-and-crazy-mandelbrot-set-full-of-energy-matthias-hauser.html.
“[Reversed] Fantasy Land - Mandelbrot Fractal Zoom Out (4k 60fps).” YouTube, YouTube, 14 Nov. 2018, www.youtube.com/watch?v=zk1H1DdWFC4.
Let’s play with this diagram and look at the bigger heart-shaped circle and graph the numbers. As seen, when we plug -0.57+0.2i or any other number from this cardioid-shaped region, the iterations are being sucked into a specific number. The blue points represent the iterations. You can look at this diagram from this link: https://www.geogebra.org/m/mfewjrek. Any number from this region will be sucked into a single point.
However, to the left of the main bigger cardioid, attached to it at the point -0.75, a circular-shaped bulb is visible. This bulb consists of those parameters c which has 2 attracting points as opposed to 1. For example, as we discussed, -1 which oscillates between -1 and 0 is a member of this bulb. There are many bulbs attached to the Mandelbrot set and each of them has a different number of attractive points. By attractive points, we mean where particles are prone to come to a stop. They are the points that our set oscillates between. Thus, in addition to displaying if a number is stable, the Mandelbrot set also shows how stable a specific set of numbers is. The picture on the right shows how many attractive points the regions have. Lastly, when we zoom into these little bulbs, we see new Mandelbrot sets which are also very interesting. This phenomenon is displayed in various youtube videos.
