Pretty pictures in the complex plane

Emily Clapham

18 October 2017

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Some of the greatest works of art in history have been produced by mathematicians. One fascinating source of mathematical artwork is fractals: infinitely complex shapes, with similar patterns at different scales. Fractal geometry has dramatically altered how we see the world. Technology has many uses for fractals, one of which is the production of beautiful computer graphics. These pretty pictures are used to present a large amount of information about a function in a clear and comprehensible manner,  and the simplicity of the maths involved in producing these pictures is fascinating.

Modern art studies have often been dismissive of the power of beauty in mathematics, with the idea that “beauty is not in itself sufficient to create a work of art”. Mathematics produces rigid and inflexible answers, whereas art is free-moving and open to interpretation. However, it is undeniable that these pretty pictures demonstrate true beauty, not only in the images but also in the mathematics behind them.

The mathematics behind pretty pictures

Extremely simple functions can be used to produce these pictures. For example, let’s consider the quadratic function $f\hspace{0.4mm}(x\hspace{0.3mm})=x\hspace{0.3mm}^2+c$, for some constant $c$. An iterative method is applied to the function. First, a seed (let’s call it $x_0$) is selected to be the initial value for iteration. The solution of the function is then subsequently recycled as the new input value, $x$. In this way:

\begin{align*}
x\hspace{0.3mm}_1&=x\hspace{0.3mm}_{0}^2 + c,\\
x\hspace{0.3mm}_2&=x\hspace{0.3mm}_{1}^2 + c = (x\hspace{0.3mm}_{0}^2 + c)^2 +c,\\
x\hspace{0.3mm}_3&=x\hspace{0.3mm}_{2}^2 + c = \cdots\\
\text{and in general, }x\hspace{0.3mm}_n&=x\hspace{0.3mm}_{n-1}^2+c.
\end{align*}

We continue until the iteration either converges to a fixed point or cycle, or diverges to infinity. The orbit is the sequence of numbers generated during the process of iteration: $x_0,x_1,x_2,x_3,\ldots,x_n$. If we only apply real numbers to the quadratic function we limit the graphical representation of the iterations to a line. To produce pictures in the plane, we use complex numbers instead.

Through the process of iteration, each seed will either converge or diverge, and so for a given function we can divide the plane into an escaping set $E_c =\{ z_0 : |z_n| \rightarrow \infty \, \mbox{as} \, n \rightarrow \infty \}$ (that is, all the seeds that end up at infinity) and prisoner set, where the iteration tends to a point or becomes periodic.

The Julia set of a function

To go from the iterative procedure described above to the vivid images to the right, we need to introduce the idea of the Julia set of a function, named after the French mathematician Gaston Julia. Julia was an extraordinary man, who tragically lost his nose while fighting in the first world war. Despite the substantial injury, he made immense progress in the field of complex iteration and published the book Mémoire sur l’itération des fonctions rationnelles in 1918, which began the study of what we now call a Julia set.

The filled-in Julia set is the collection of points in the complex plane that form the prisoner set of a function, while the Julia set itself is the boundary of this region. The points within the filled-in Julia set remain bounded under the iteration since their orbits converge to an attracting point or cycle.

Connected and unconnected Julia sets of the quadratic function for different values of $c$

Conventionally, when pictures of the Julia set are shown, the filled-in Julia set is shaded black and varying colours are used to show the rate at which the escaping set diverges to infinity. The Julia set is therefore the edge of the black region. Maps 1–7 above show the Julia sets of the quadratic function for different values of $c$, with the escaping set colour-coded as follows: red areas represent points that slowly escape from the set, while blue areas signify points that quickly escape to infinity. The value of the complex constant $c$ influences the shape of the Julia set.

Maps 1, 4 and 5 all have black centres, which indicate that the Julia set is connected, while maps 3, 6 and 7 demonstrate unconnected sets. For these images, the Julia sets have no black regions and instead the pictures are just flurries of colour. It is not always easy to spot whether a Julia set is connected, however. In map 2, there is no obvious black region, but neither are there colourful individual flurries and instead we see a spiky line. In fact the set is connected, it is just that the filled in Julia set is so slender that the black line points are not visible in the image.

During the initial study of these sets, a fascinating criterion for connectivity was discovered concerning the critical point, $z_0=0$. If the critical point is used as the seed, we produce the critical orbit, which is bounded if and only if the Julia set is connected.

Fractal patterns appear in all plots, apart from when $c=0$ or $-2$. The picture below displays examples of magnified sections of the fractals, for $c=-0.7$ (maps 9–12), $c=-0.12 + 0.75 \,\mathrm{i}$ (maps 13–16), $c=0.1 + 0.7 \, \mathrm{i} $ (maps 17–20) and $c=-0.1 + 1 \, \mathrm{i} $ (maps 21–24). Each enhancement of a section produces what appears to be copies of the whole section, not just in overall shape but also with smaller embellishments on every “limb”. For connected plots, these fractals appear as loopy ovals and circles or thin, almost stick-like, sections. For disconnected plots, however, the fractals are grouped together in intricate floral patterns, revealing the same shape and pattern with each level of magnification.

Magnified sections of fractals for different values of $c$

Prior to computer technology, Julia had to rely on his imagination and manually carry out the iterations by hand. Fifty years later, another mathematician applied modern computing power to plot these pretty pictures, finally showing the sets in all their beauty…

The Mandelbrot set

The Mandelbrot set is named after the Polish mathematician Benoit B Mandelbrot, known for being the founder of fractal geometry. The word fractal is derived from the Latin fractus, which means broken, and describes the shape of a stone after it has been smashed.

Mandelbrot discovered that fractals appear not only in mathematics but also in nature, through crystal formation, the growth of plants and landscapes, as well as in the structure of the human body. In 1945, Mandelbrot read Julia’s 1918 book. He was fascinated and, with the aid of computer graphics, was able to show that Julia’s work contained some of the most beautiful fractals known today.

To create the Mandelbrot set, each complex value of $c$ is used as the constant term in the quadratic function $f\hspace{0.3mm}(z\hspace{0.2mm})=z\hspace{0.3mm}^2+c$ and iterated with the critical point $z_0=0$ as the seed. If the orbit escapes to infinity, the number of iterations taken for the modulus of the function to exceed a specified value is used to decide on the colour of the map at that point, $c$. Otherwise, when the orbit converges, the point is coloured black. The Mandelbrot set is the set of black points.

For example, if we let $c=-0.15+0.3 \, \mathrm{i}$ then we have the complex quadratic function $f\hspace{0.3mm}(z\hspace{0.2mm})=z\hspace{0.3mm}^2-0.15+0.3 \, \mathrm{i}$. We start with $z_0=0$ as the seed and the sequence of iteration (to 5 significant figures) is as follows:

\begin{align*}
z_1&={0}^2 -0.15 +0.3 \, \mathrm{i} &&\Rightarrow &z_1&= -0.15 +0.3 \, \mathrm{i},\\
z_2&=(-0.15+0.3 \, \mathrm{i})^2-0.15+0.3 \, \mathrm{i} &&\Rightarrow &z_2&= 0.2175 +0.21 \, \mathrm{i},\\
&&&&z_3&=-0.14679+0.20865 \, \mathrm{i},\\
&&&&z_4&=-0.17199+0.23874 \, \mathrm{i},\\
&&&&z_5&=-0.17742+0.21788 \, \mathrm{i}.
\end{align*}

Continuing to 30 iterations, the orbit has not escaped to infinity and instead converges to the point   $z=-0.17082+0.22361\, \mathrm{i}$ (again to 5 significant figures). Therefore, $c=-0.15+0.3 \, \mathrm{i}$ is within the Mandelbrot set and is coloured black.

On the other hand, if we take $c=-1.85+1.2 \, \mathrm{i}$, and hence the complex quadratic function $f\hspace{0.3mm}(z\hspace{0.2mm})=z\hspace{0.4mm}^2-1.85+1.2 \, \mathrm{i}$, then the sequence of iterations (to 5 sf) is as follows:

\begin{align*}
z_1&={0}^2 -1.85 +1.2 \, \mathrm{i} &&\Rightarrow &z_1&= -1.85 +1.2 \, \mathrm{i},\\
z_2&=(-1.85 +1.2 \, \mathrm{i})^2-1.85 +1.2 \, \mathrm{i} &&\Rightarrow &z_2&= 0.1325 -3.24 \, \mathrm{i},\\
&&&&z_3&= -12.33+0.3414 \, \mathrm{i},\\
&&&&z_4&= 150.06 – 7.2189 \, \mathrm{i},\\
&&&&z_5&= 22465 – 2165.4 \, \mathrm{i}.
\end{align*}

The characteristic segments of the Mandelbrot set

If the modulus of $z$ exceeds 100, then it has been proven that the orbit escapes to infinity. This occurs on the fourth iteration, so the colour chosen to represent the value of 4 would be plotted at the point $(-1.85,1.2)$ in the complex plane. The resulting image is shown in map 8, and also in the picture to the left.

The largest segment of the set is called the cardioid due to its heart-like shape. Attached to this are adornments called bulbs, upon closer inspection of which it is possible to see many smaller, somewhat similar, embellishments. The bulbs are not completely identical, although most exhibit a similar shape, and the main differences can be seen in their filaments. The filaments are the thin strings of bounded points that sprout like sticks from the tops of the bulbs. These sticks are extremely narrow and they appear to be coloured red, which would indicate they are not part of the set. However, if we were to zoom in closer on these regions, we would actually see black lines!

The self-similarity of the Mandelbrot set

The Mandelbrot set is self-similar, consisting of miniature Mandelbrot sets within the boundary of the largest set. By enhancing the filaments, smaller copies of the overall set appear in ‘Russian-doll’ like fashion, as seen in maps 26–30 above. Closer inspection of map 27 shows many more self-similar sets within the filaments around the perimeter of the Mandelbrot set. Magnifying the small copies of these Mandelbrot sets would yield infinite layers of self-similar sets.

Other fascinating and intricate shapes occur, for example the “seahorse valley” that is visible in maps 31–34 above. By enhancing the plot within this region we see two rows of seahorse shaped embellishments, each with “eyes” and “tails”. Further magnification of the “eyes” reveals spiral constellations of more “seahorses”.

Connection between Julia sets and the Mandelbrot set

The orbit of the critical point $z_0=0$ can be used to test the connectivity of the Julia set, and the Mandelbrot set shows the boundedness of these critical orbits. Hence, the Mandelbrot set itself indicates the connectivity of the Julia sets of all the different complex quadratics. The Mandelbrot set can be described as $M = \{ c \in \mathbb{C} \, | \, J_c \, \mbox{is connected}\} $, where $J_c$ is the Julia set of the function $z\hspace{0.3mm}^2+c$. The Julia set is a connected structure if $c$ is within the Mandelbrot set, and will be broken into an infinite number of pieces if $c$ lies outside the Mandelbrot set.

The cardioid-shaped main body contains all values of $c$ for which the Julia set is roughly a deformed circle (figure below: maps 35, 37, 38 and 40). The values of $c$ which lie in a bulb of the Mandelbrot set produce a Julia set consisting of multiple deformed circles surrounding the points of a periodic attractor. The number of subsections sprouting from a point on the Julia set is equal to the period of the bulb in the Mandelbrot set (below; maps 36, 39, 41–44).

A specific Julia set can be defined by a point in the Mandelbrot set

The nature of the convergence of points within the Mandelbrot set depends on the segment in which the point resides. Seeds within the cardioid converge to an attractive point, whereas orbits starting in the bulb lead to an attracting cycle.

Three particularly interesting cases of Julia sets are shown below. The first is when $c=0$, where the filled-in Julia set comprises of all the values within the unit circle (circle of radius 1, centred on the origin) and each of these points converges to $0$ when iterated. The Julia set is the boundary of the circle, the points of which, when iterated, remain on the boundary.

Three remarkable examples of the Julia set with $c=0$, $c=\mathrm{i}$ and $c=-2$

The second interesting case is when $c=\mathrm{i}$. Here, the Julia set is a dendrite, meaning there are no interior points. Instead, the set is just a branch of points. For this complex constant the dendrite is a single line in an almost lightning-bolt shape. The final case is $c=-2$, where the Julia set is a dendrite that lies directly on the horizontal axis between  $-2<x<2$.

Explore the sets yourself

I hope to have displayed the beauty behind these pictures by emphasising the extraordinary quantity of information contained in such a simple procedure, as well as through highlighting the complexity of each image, in the variety of fractals and colours visible, which further enhances the beauty.

If this article has sparked an interest in fractals, then why not try exploring these sets for yourself? You could do this by magnifying different sections of the Mandelbrot set to explore the countless shapes and patterns that exist within. You could also go deeper into exploring individual orbits.

All of these pictures are generated using simple quadratic formula. However, the Julia and Mandelbrot sets can be produced for a wide variety of functions in a similar manner to obtain countless pretty pictures.

These images are already becoming dated, having been taken for granted for so many years since they were first produced on the big bulky computers of the 1980s. The Julia set of the quadratic function, and the corresponding Mandelbrot set, could be inspiration for pretty pictures which are yet to be fully explored, or even discovered. Largely, the discoveries discussed here have been recorded in recent years. Furthermore, there could still be vast amounts of information within these sets that are yet to be discovered. Could you be the one to make a discovery?

References

Emily Clapham

Emily is a mathematics graduate from Sheffield Hallam University. She is taking a year out to write GCSE study revision guides, before pursuing a career in teaching. Emily is hoping to interest more children into taking STEM subjects.

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