What Is The Mandelbrot Set And Where Did It Come From?

Ask any random person for an example of a fractal, and there are a few answers you can expect. The first, obviously, is “Who are you? Stop asking me math questions lady, this is a Wendy’s”. The second, though, may well be this:

It’s called the Mandelbrot – or more properly, Mandelbröt – set, and it’s probably one of the most famous pieces of math in the world. That’s partly thanks to its psychedelic visuals – but focusing on just that means ignoring the important story of how it was discovered.

So what actually is the Mandelbrot set? And where did it even come from?

What is a fractal?

At its core, the Mandelbrot set is a fractal – so let’s make sure we understand what that means.

There are two ways to define fractals: the fuzzy way, which is pretty easy to understand, and the mathematically rigorous way, which is not.

“They are tricky to define precisely,” explained Michael Rose, then a PhD candidate in the University of Newcastle’s School of Mathematical and Physical Sciences, in a 2012 article for The Conversation, “though most are linked by a set of four common fractal features: infinite intricacy, zoom symmetry, complexity from simplicity and fractional dimensions.”

In other words, a fractal is a complex shape produced from very simple rules; one which, no matter how far you “zoom in”, will never get any simpler nor more smooth than how it started; and one whose dimension sits somewhere in between two integers, like this. 

They’re barely contained by strict mathematical definition, but the easy description is this: they’re “beautiful, damn hard, increasingly useful. That’s fractals.” 

And for both explanations, we have one person to thank perhaps more than anyone else: Benoît Mandelbröt.

Who was Benoît Mandelbröt?

Born in Warsaw, Poland, in 1924, the improbably named Benoît B Mandelbröt – the ”B” stands for a non-existent middle name – would be so important to the story of fractal geometry that he pretty literally gave his name to the field.

“Benoit was a Renaissance man who created a renaissance,” wrote Nathan Cohen in the 2015 book Benoit Mandelbrot: A Life In Many Dimensions. 

“Mandelbrot could ‘see’ answers to important mathematical questions […] defining an entire field of geometry – fractal geometry,” Cohen wrote. “Then, just to thrust home the point, he commanded – he saw – that fractals are the geometry of nature. And he saw this with great clarity. Note the absence of ‘a’ or ‘one of’. He left no doubt to that interpretation.”

Given his fame today as a mathematician and his own preoccupation with the beauty of nature, Mandelbröt was perhaps not who you might think in his everyday life – he was actually kind of a code monkey, working at IBM in their Yorktown Heights research center in New York. That was on purpose: back in the 50s and 60s, taking a post like this afforded Mandelbröt more freedom to pursue the sprawling, apparently disparate research topics he was interested in, rather than following some hyper-specific academic project.

“I realized that mathematics cut off from the mysteries of the real world was not for me, so I took a different path,” he would later write in his memoir, The Fractalist: Memoir of a Scientific Maverick. He was, he said, interested more in “questions once reserved for poets and children.”

So, it was at IBM in the 1970s that Mandelbröt rediscovered a paper he had first come across as a fresh-faced 21-year-old: Mémoire sur l’iteration des fonctions rationnelles, by Gaston Julia.

The first fractals

Mandelbröt may be the most famous name associated with fractals, but he wasn’t the first person to discover them. In fact, for a concept so ubiquitous throughout the world, the discovery of fractals was actually a long and painstaking process, with each new development coming sometimes decades after the previous work that made it possible.

The problem with fractals, for a long time, was that first step: leaving behind differentiability. Intuitively, a function is “differentiable” if its graph looks… well, if it looks nice and smooth, really – no pointy bits, or breaks, or shooting off to infinity halfway through. 

“Until the 19th century, mathematics had concerned itself only with functions that produced differentiable curves,” explained Holly Trochet, then a pure mathematician at the University of St Andrews, in 2009. “However, on July 18, 1872, Karl Weierstrass presented a paper at the Royal Prussian Academy of Sciences showing […] the first rigorously proven example of a function that is analytic, but not differentiable.”

Take a look at the graph of that function, and you can maybe see why people had avoided this kind of thing until then: it’s messy, pointy, and seemingly unpredictable. It “resisted traditional analysis,” Trochet wrote; functions of this type were “labelled ‘monsters’ by Charles Hermite and […] largely ignored by the contemporary mathematical community.”

But once that dam had been broken, it was only a matter of time before the world would be swimming in fractals. In 1883, Georg Cantor introduced his Cantor set – the word for it was yet to be coined, but this would later be recognized as one of the very first fractals defined in math. A couple of decades later, still riffing on Weierstrass, Helge von Koch created the Koch curve and snowflake – also fractals, though again, he wouldn’t have thought of them as such.

It was in the late 1910s, though, that three giants of fractal geometry turned up – people without whom Mandelbröt would just be some particularly eccentric IBM dude. First, Felix Hausdorff, who in 1918 introduced the concept of the Hausdorff dimension – without that, we can’t fulfill the “non-integer dimension” part of the definition of fractal. Second, a pair of French mathematicians with one nose between them: Pierre Fatou and Gaston Julia – the same Julia whose paper would later inspire Mandelbröt.

From Julia to Mandelbröt

So why were these two French guys so influential? Well, it’s pretty fair to say that without them – and Julia in particular – there would be no Mandelbröt set at all. 

So, what did Julia even do? Well, he invented the Julia set. Defined loosely, it’s the set of points where, no matter how many times you repeatedly apply some function to them, they will never shoot off to infinity. That doesn’t sound all that interesting on its own – a set of points that doesn’t do something weird – which may be why the idea fell into obscurity for so long.

But fast forward to the computer age, and the Julia set… well, it gets really special.

“With the aid of computer graphics, Mandelbrot […] was able to show how Julia’s work is a source of some of the most beautiful fractals known today,” wrote John O’Connor and Edmund Robertson, both pure mathematicians at the University of St Andrews, in 1999. “To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.”

But it was another spark of inspiration that would lead to the iconic Mandelbröt set: rather than focus on the values z under that iteration, he decided to map the values c for which the Julia set for the function f_c(z)=z² + c is connected – that is, all one big thing.

The result: the Mandelbrot set. 

“The Mandelbrot set is, for many, the quintessential fractal,” Trochet wrote. “When one zooms in on some part of the edge, one notices that the Mandelbrot set is, indeed, self-similar.”

But it wouldn’t have been possible without those who came before him. Even the fractal pattern that bears his name is created from Julia sets: indeed, focus in on any boundary point of the Mandelbröt set, and it’s functionally the same as a Julia set.

“While the lion’s share of the credit for the development of fractal geometry goes to Benoît Mandelbröt, many other mathematicians in the century preceding him had laid the foundations for his work,” Trochet wrote. “However, this in no way detracts from his visionary achievements.”

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