History & the people

The story starts with a crisis of confidence in calculus. In the 19th century mathematicians were rebuilding the subject on careful definitions, and the careful definitions kept admitting objects nobody had imagined. In 1872 Karl Weierstrass showed the Berlin Academy a function, built as an infinite sum of ever-faster cosine waves, that is continuous everywhere and differentiable nowhere. Its graph is an unbroken line with no direction at any point.

Georg Cantor, working on the theory of infinite sets, published his "middle thirds" set in 1883: a dust of points with no total length that still has as many points as the whole line. In 1890 Giuseppe Peano produced a curve that passes through every point of a square, and in 1891 David Hilbert gave a simpler one, with a drawing. In 1904 Helge von Koch built his snowflake curve with ruler and compass, to show Weierstrass's phenomenon in plain geometry. Wacław Sierpiński described his triangle in 1915 and his carpet in 1916; Karl Menger built the sponge in 1926.

None of these people were trying to describe trees or coastlines. Each shape was a counterexample or a tool in a specific argument: about continuity, about dimension, about what a "curve" is. Many of their colleagues regarded the shapes as pathological, and they were sometimes called monsters. Chapter 3 builds each one and measures it.

The other root of the subject is iteration: take a function, apply it to a number, apply it again to the answer, and keep going. For a function as simple as z² + c, where z and c are complex numbers, the question "what happens in the long run?" turns out to have extraordinarily complicated answers. Some starting points fly off to infinity. Some settle down. The boundary between the two behaviors is what we now call a Julia set.

Two French mathematicians, Pierre Fatou and Gaston Julia, built the theory of these iterations in a burst of work between 1917 and 1919. The French Academy of Sciences had set iteration of rational functions as the subject of its 1918 Grand Prix, and Julia's long memoir won it. Julia did this work after severe injury in the First World War, in which he lost his nose; he wore a leather strap across his face for the rest of his life.

What Fatou and Julia did not have was a picture. Every result was derived by pure reasoning about infinitely many iterations. They knew the boundary sets could be extraordinarily intricate, but there was no way to draw one in any detail by hand. Their work was admired and then largely set aside for half a century, until computers made the question visual. The Julia sets chapter shows what they were reasoning about.

Lewis Fry Richardson was an English physicist and pacifist with two unusual research programs. In 1922 he published Weather Prediction by Numerical Process, the first serious attempt to forecast weather by solving the equations of the atmosphere step by step, decades before there were computers to do the arithmetic. Later, studying what makes nations go to war, he wanted to know whether countries that share longer borders fight more often. That needed border lengths, and he found that the published figures disagreed, sometimes badly, between neighboring countries.

The reason was the ruler. Measure a coastline by walking a pair of dividers along a map, and a smaller divider setting follows more of the wiggles and gives a longer answer. For a smooth curve the answers converge; for a coastline they keep growing. Richardson plotted measured length against ruler size on logarithmic axes and found straight lines, a power law, with a different slope for each coast. He died in 1953; the measurements were published in 1961.

In 1967 Benoit Mandelbrot turned Richardson's slopes into a geometric idea in a short paper in Science, "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension". A coastline has no well-defined length, he argued, but it does have a well-defined dimension: about 1.25 for the west coast of Britain, by Richardson's data. It was the Koch curve's idea, roughened up by chance.

Meanwhile the iteration theme returned from an unexpected direction. In 1963 the meteorologist Edward Lorenz, at MIT, published "Deterministic Nonperiodic Flow". He had found that a simple model of convection, three equations, never settles down and never repeats, and that a tiny change in the starting numbers grows until the forecast is completely different. He noticed it when he restarted a simulation from numbers printed to three decimal places instead of the six the computer held. The orbit of his model traces a shape now called the Lorenz attractor, a strange attractor whose cross-sections are fractal. Chapter 11 runs it.

In 1968 Aristid Lindenmayer, a Hungarian-born theoretical biologist, published a formal grammar for the growth of simple multicellular organisms such as algae: rewrite every symbol in a string by fixed rules, in parallel, generation after generation. These L-systems later became the standard way to grow plants on computers, and many classic fractal curves are one-line L-systems. Chapter 5 covers them.

And in 1976 the ecologist Robert May published "Simple mathematical models with very complicated dynamics" in Nature, showing that the logistic map, x → r x (1 − x), a toy model of population growth, passes through period doubling into chaos as r increases. Mitchell Feigenbaum, at Los Alamos, had been computing the same cascade on a programmable pocket calculator and noticed that the gaps between doublings shrink by the same ratio, about 4.669201609, in every such system. He published this universality result in 1978. The bifurcation diagram, with its smaller and smaller copies of itself, is one of the most recognizable fractals there is.

Benoit Mandelbrot was born in Warsaw in 1924, grew up in France, and in 1958 joined IBM's research division in New York, where he stayed for most of his career. IBM gave him two rare things: freedom to follow a strange idea across fields, and computers. His early work looked unrelated: price swings in cotton markets, bursts of noise on telephone lines, the flooding of the Nile, word frequencies. In each he found the same thing, irregularity that looks alike at every scale.

In 1975 he gave the idea a name. His French book Les objets fractals: forme, hasard et dimension introduced the word fractal, from the Latin adjective fractus, broken, from the verb frangere, to break. An English version, Fractals: Form, Chance and Dimension, followed in 1977, and in 1982 the much expanded The Fractal Geometry of Nature, the book that took the word into everyday language. Its central claim was that clouds are not spheres and mountains are not cones, and that the monsters of 1872–1926 were the right geometry for nature after all.

Around 1979 and 1980 Mandelbrot turned to the old work of Fatou and Julia, now with computers. Rather than draw one Julia set at a time, he mapped the parameter c itself: for which values of c does the iteration of z² + c, started from zero, stay bounded? The set of such c is what we call the Mandelbrot set. He published pictures of it in 1980. They were coarse, but the shape was there: the cardioid, the circular bulbs, and specks that looked like separate islands.

He was not quite the first to print it. Robert Brooks and Peter Matelski, studying groups of transformations in work presented at a 1978 conference, included a crude computer printout of the same set, drawn in asterisks. Their interest was elsewhere, and the picture drew little notice at the time.

The mathematics came quickly after the pictures. In 1982 Adrien Douady and John Hubbard proved that the Mandelbrot set is connected: the apparent islands are all joined to the main body by filaments too thin to show at that resolution. They also gave the set its name. Their work began a flood of research in complex dynamics, which is still going on.

At SIGGRAPH 1980, the computer graphics conference, Loren Carpenter, then an engineer at Boeing, showed a short film called Vol Libre: a flight through mountains generated by repeatedly subdividing triangles and nudging the new points up or down at random, with smaller nudges at finer scales. It was a striking early demonstration that fractal rules could make realistic landscapes, and it got him hired by Lucasfilm's computer graphics division.

Two years later that group made the Genesis sequence in Star Trek II: The Wrath of Khan (1982), a fully computer-generated minute in which a dead planet is engulfed in fire and transformed into a living world, with fractal mountains. It was one of the first entirely computer-generated sequences in a feature film. The same year Carpenter, Alain Fournier and Don Fussell published the method, "Computer Rendering of Stochastic Models", in Communications of the ACM. Lucasfilm's graphics group later became Pixar.

Mathematics was supplying new tools at the same time. In 1981 John Hutchinson showed that any finite set of contracting maps has exactly one shape that is a union of shrunken copies of itself, and that iterating the maps from any start converges to it. Michael Barnsley built on this with Stephen Demko in 1985 and in his 1988 book Fractals Everywhere, introducing the "chaos game" way of drawing such shapes and the famous fern drawn by just four affine maps. Also in 1981, the physicists Thomas Witten and Leonard Sander described diffusion-limited aggregation, a random walk process that grows branching clusters like frost, soot and mineral dendrites.

One more curve deserves its place here. The Heighway dragon was found in the 1960s by the NASA physicists John Heighway, Bruce Banks and William Harter, by folding a strip of paper in half again and again and opening it out to right angles. Martin Gardner introduced it to the public in his Scientific American column in 1967. Its successive folds also decorate the section openings of Michael Crichton's novel Jurassic Park.

By the late 1980s fractal programs were everywhere personal computers were, and new families kept arriving. In 1989 Mario Markus published pictures of what are now called Markus–Lyapunov fractals: maps of where a population model alternating between two growth rates is stable or chaotic, colored by its Lyapunov exponent. In 1992 Michael Michelitsch and Otto Rössler described the Burning Ship, a one-line change to the Mandelbrot formula that takes absolute values before squaring, and produces a shape that looks like a ship in flames.

The mathematics kept deepening. In 1998 Mitsuhiro Shishikura proved that the boundary of the Mandelbrot set has Hausdorff dimension exactly 2: the edge is so convoluted that, in the sense of dimension, it is as big as a surface. Whether it has positive area is still an open question.

Computer artists pushed the look. Scott Draves had been developing a way of rendering iterated function systems with nonlinear "variations", per-point color and logarithmic density; he and Erik Reckase described it in "The Fractal Flame Algorithm" (2003, revised 2008). Flames became the basis of his long-running Electric Sheep screensaver and of a generation of fractal art tools. Chapter 6 explains the algorithm.

The long-sought three-dimensional Mandelbrot set came in 2009, when Daniel White and Paul Nylander settled on a formula using spherical coordinates raised to the eighth power, the Mandelbulb. A year later Tom Lowe introduced the Mandelbox, built from box folds and sphere folds instead of squaring. Both are rendered by marching rays toward a distance estimate, which Chapter 14 describes.

And deep zooms became practical. In 2013 K. I. Martin described a perturbation method: compute one reference orbit at high precision, then compute every other pixel as a small difference from it in ordinary floating point. Zooms that would take weeks of arbitrary-precision arithmetic now take minutes. Color work moved forward too: Björn Ottosson's OKLab color space (2020) made smooth, even-looking gradients easy, and is now used for fractal palettes, including the 180 schemes in JellyDazzle, the audio-reactive kaleidoscope app this site grew alongside.