There's an entire world out there – one rich in valleys, hills, parading elephants, and infinite detail – that can be described in just four words. It may sound fantastical, but not only is this world is very real – it also hides a mystery that the world’s top mathematicians have been grappling with for nearly half a century.
Those four words? z squared plus c.
Those four words might not make sense right now, but soon you will see how such a short phrase can unlock multitudes and infinities. Welcome to the world of the Mandelbrot set.
At a first glance, the Mandelbrot set looks a bit like an ink blot test. But when you ‘zoom in’, in the same way you would zoom into a photograph, its psychedelic nature reveals itself.
Circles and spirals spawn and swirl, like a relentlessly hypnotic naughties-era screensaver, often coloured in garish neon to amuse a still-thriving corner of YouTube. Perhaps the most amazing thing is that, no matter how much you zoom, it never gets any less detailed.
The Mandelbrot set has long been a fascination of scientists and artists alike. Essentially, it’s a visualisation of how an equation, those four magic words (z squared plus c), affects different numbers.
It was first defined in 1978 by Prof Benoit Mandelbrot, and mathematicians have been trying to work it out ever since.
In fact, in 1985, Prof John Hubbard, mathematician at Cornell University in the US, authored a paper showing that the Mandelbrot set is one connected piece – but that very same paper posed questions that, even 45 years later, we still have no answer to.
Some mathematicians (including Hubbard himself) have been working on solving these problems for their entire careers.
Meanwhile, in an unusual turn of events for mathematical equations, the Mandelbrot Set broke into mainstream culture. Visualisations were printed off or torn from magazines and pasted on the walls of university students and maths teachers alike.
Arthur C Clarke referenced it in his 1990 novel, The Ghost from the Grand Banks. Even the performance artists Blue Man Group got in on the action, referencing Mandelbrot in the titles of three songs in their debut album The Complex.
But, while the world has moved on from the 80s, the Mandelbrot set has not. So how close are we to figuring it out?
Complexity from simplicity
Nothing in life is ever as simple as it seems. A snail slithers across the path in front of you, and you might notice its spiral shell. You might (if you’re not averse to slime) pick it up and study the spiral. It looks relatively straightforward – pretty, but otherwise uninteresting.
But this spiral follows a mathematical pattern, and can be mapped with an equation that leads to what is, arguably, a piece of art that needs to be seen to be appreciated: the Fibonacci sequence.
In the same way, our four words (z squared plus c) lead to an intricate pattern, far beyond what you might expect from the words themselves.
The real beauty of these four words is that they are so easy to state, and yet lead to the Mandelbrot set which, visually, is so complicated. As Associate Prof Anna Benini, from the University of Parma in Italy, wrote in a 2017 paper, the equation z squared plus c, “despite its simple (apparently innocent) form, exhibits a rich variety of dynamical behaviour.”
So what do these four words actually mean, and how does this lead to the Mandelbrot set? Well, this is where it gets a little bit complicated.
We start by saying z is 0, and our chosen number is c. Then we run those four words: z squared plus c. The next step is to take the result of that, make this answer the new z and run it all again. Then, we take the result of that and make that our new z and keep going. This process of casting this spell again and again and again is called iteration.
The Mandelbrot set is the collection of ‘c’s that, no matter how many iterations we apply, yield answers that are no more than 2 away from 0.
So far, so good – right? But (there’s always a but) the Mandelbrot set is two dimensional, so we cannot use ‘normal numbers’ – the numbers you’ve probably been familiar with so far, like 1 or -½. Instead, we’re talking here about complex numbers. Don’t panic: these may sound intimidating, but complex numbers are not as scary as they seem.
When ancient mathematicians came across the problem of subtracting from zero, they began using negative numbers. Complex numbers emerged to answer the question: “What is the square root of a negative number?”
Enter i, defined as the number that, when squared, gives -1. Any multiple of i is called an ‘imaginary number’, for example 2i or -5i. Add an imaginary number to a real number (numbers that are not imaginary) and you get a complex number. Voilà!
To visualise this, we can plot complex numbers on a graph, with the real part on the horizontal axis, and the imaginary part on the vertical axis.
It is on this graph that the Mandelbrot set comes to life.
Remember that we defined the Mandelbrot set as the collection of ‘c’s that are never more than two away from 0? This can be shown by drawing a circle on our graph, with the radius limited to 2. The Mandelbrot set are the values of c which never leave the circle as we iterate.
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Seeing infinity
“The Mandelbrot set is a model of what to anticipate in much more general and complicated situations,” says Prof Mikhail Lyubich, a world-leading researcher in the Mandelbrot Set at Stony Brook University in the US.
In other words, if we can fully understand the Mandelbrot set, we can then use what we have learned to explain observations that we can see around us.
Even in maths alone, the Mandelbrot set can be used as a simplified version of situations that arise in some of the most talked about areas of maths, from quantum field theory to hyperbolic geometry.
These fields are as complicated as they sound, so much so that even world-leading mathematicians didn’t immediately realise the intricacies of the Mandelbrot within them.
Hubbard, the mathematician who’s been working on the set since his 1985 paper, says that its visual results are more than just pretty pictures. In fact, it was only when he showed them to leading mathematician Prof Lars Ahlfors that Ahlfors understood their true significance.
“If even Ahlfors couldn’t figure out why it was interesting without seeing these pictures, it shows that those pictures are not just hype,” he says. “They're absolutely essential to the understanding.”
Yes, it would be possible to work with just the four words, but without seeing the Mandelbrot set, we don’t know what to expect.
Because the Mandelbrot set is a graph, showing a set of numbers (those who have been granted entry to the club), then we can zoom in on any section of the graph and inspect which numbers are in our ‘gang’ more closely.
The never-ending rows of ‘elephants’ and ‘seahorses’ (common shapes found within the set) demonstrate another aspect of the Mandelbrot set that has so enthralled people: its infinite nature. The Mandelbrot set is comprised of an infinite number of intricate patterns, that remain just as detailed on magnification.
There is another aspect in which the Mandelbrot set is infinite too, and one that can’t just be seen visually. Look at the bulb on the left-hand side of the set. You might see that, moving left, there are lots of similar bulb-like shapes. These are, in fact, exact copies of the original bulb.
But how can something contain an exact copy of itself? And because the Mandelbrot set is infinite, each bulb can contain an infinite number of bulbs. It’s like an infinite number of Russian Dolls… which are all the same size.
This Russian doll property is called ‘self-similarity’ and it was first demonstrated in 1990. Because we know that the Mandelbrot set is self-similar, then there will always be another, smaller bulb to the left of whichever bulb you are looking at.
The final obstacle
There’s a lot we know about the Mandelbrot set – but one key debate, first posed in 1985, is still unsettled. This is the holy grail of Mandelbrot set mathematics. This is MLC.
MLC stands for Mandelbrot Locally Connected, and essentially describes how different parts of the Mandelbrot set (the valleys of elephants or seahorses, for example, or the endlessly repeating bulbs) are connected to each other.
Loosely defined, a set is ‘connected’ if you cannot split it up into non-overlapping regions (that do not include their edges). A quick test to confirm a set is connected: if you can put your pen on any given point and draw a line to any other point without leaving the set, then the set is connected.
‘Local connectivity’ is a version of this that looks at a very small region around a point (which, in the Mandelbrot set, is a complex number). Take any point in the set and draw a little ‘neighbourhood’ around it.
This neighbourhood could be circular, or square, or even just a blob. Now try to draw an even smaller neighbour within the outer neighbourhood (let’s call it a sub-neighbourhood) that both contains our point and is connected.
Still following? We say the point is ‘locally connected’ if we can find such a sub-neighbourhood, regardless of what the outer neighbourhood is. A set is locally connected if every point in it is locally connected.
If your head is hurting, imagine a fine-toothed hair comb. But this is not any old comb. We start with just one tooth, in the middle. Then we place a tooth ¼ of the way along from the left-hand side. Then another ⅛ of the way along and so on, each time placing a new tooth midway between the left-hand side, and the left-most point.
We do this an infinite number of times. This is definitely connected – it is all part of the same comb. If you pick the comb up, it doesn’t matter where you put your fingers, the whole comb will move together. So we say the comb is globally connected.
However, rather confusingly, it’s not considered locally connected. Pick a point on the left side. Our outer neighbourhood can be a little circle, small enough that it doesn’t include the handle or spine of the comb.
Now, because we know that there are teeth getting infinitely close to the left-hand edge (due to the way we made the comb), in whatever sub-neighbourhood we draw there must be another tooth. But without the spine of the comb, the teeth aren’t connected to each other. We can’t draw a connected sub-neighbourhood, so the comb isn’t locally connected.
This highlights a key fact: a set can be connected without being locally connected.
Most experts believe MLC to be true – in other words, that the Mandelbrot set is locally connected. But, so far, there has been no concrete proof of this.
Establishing MLC is crucial to our understanding of the set. As Hubbard explains, “local connectivity is almost synonymous with describable.”
So, if we know that the Mandelbrot set is locally connected, then we can tell you which points are in MLC. The flip side is a far scarier thought for Hubbard, though. “If it isn't locally connected, then there are spots in the Mandelbrot set that we do not know about.”
In the 40 years that mathematicians have been working on MLC, some progress has been made. One of the key strategies looks at special kinds of points that are known as ‘finitely renormalisable’.
Remember the Russian Dolls from before? If a point is in only one copy of the Mandelbrot set (inside only the outer Russian Doll) it is called ‘once renormalisable’. If it is in the first and second Russian Doll, but not the third, it is ‘twice renormalisable’.
Now, some points are in an infinite number of Russian Dolls. These are called ‘infinitely renormalisable’ (and anything else is ‘finitely renormalisable’).
This was the approach of Prof Jean-Christophe Yoccoz, who in 1989 showed that all finitely renormalisable points are locally connected. From this, we know that if a point is not locally connected, it must be infinitely renormalisable. But we still don’t know if such a point exists.
A timely solution
Since 1989, the proportion of points that might not be locally connected has been whittled down further and further. Between 2006 and 2009, Lyubich and his long-term collaborator Prof Jeremy Kahn proved that certain types of infinitely renormalisable points (known as decorations and molecules) are locally connected.
Even as recently as December 2023, Lyubich and Associate Prof Dzmitry Dudko conquered another type of points (these ones were called Feigenbaum points). Another swathe of potential barriers to MLC knocked out.
The problem is, though, when you start with an infinite number of points that might not be locally connected, and you manage to show that half of them are, in fact, locally connected, you are still left with an infinite number of points that might not be.
So… will we ever prove MLC? “We have set a strategy that potentially can bring us to the end of this journey in finite time, but would still require us to climb several steep peaks,” says Lyubich. “I cannot give you any definitive promises, but I am optimistic.”
Hubbard is similarly hopeful. He mentions that there are several other methods that people, including himself, are working on, each of which may lead to a proof.
Suffice to say that none of these methods are straightforward, and will involve ideas that are new even to most professors of mathematics. (And, before you mention artificial intelligence, a problem this complicated is well beyond the current capabilities of the technology.)
It truly is remarkable how something so simple to explain can produce such complicated dynamics but, as Hubbard says, maybe this is a metaphor for life. After all, the human genome only contains three gigabytes of data, and yet within that is encoded the twists and folds inside each of our brains.
Those three gigabytes encode how organs as complicated as the kidneys or our eyes are formed with all their different components. It encodes what colour our hair may be, and how tall we may grow.
Just as the Mandelbrot set contains worlds, multitudes, and infinities inside of it, so do, in many ways, each of us. Perhaps that is why we find the Mandelbrot set so beautiful.
About our experts
Prof John Hubbard is a Professor of Mathematics at Cornell University in the US. He has been published in Texts in Applied Mathematics, Journal of Differential Geometry and The American Mathematical Monthly to name a few sources.
Prof Mikhail Lyubich is a world-leading researcher in the Mandelbrot Set and Professor of Mathematics at Stony Brook University in the US. He has been published in Communications of the American Mathematical Society, Annales Scientifiques de l École Normale Supérieure and Geometric and Functional Analysis to name a few journals.
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