One line of arithmetic, repeated. Take a number c, start at zero, and apply z → z² + c over and over. If the running value stays small forever, c belongs to the Mandelbrot set; if it ever exceeds 2 in absolute value, the sequence escapes to infinity and c is out. That is the whole definition. The picture it produces has a boundary as rough as a curve can be, structure at every magnification anyone has computed, and an open problem attached to it that has resisted proof for four decades. The explorer embedded in this article draws it in real time and will zoom until the arithmetic of your browser runs out of precision.
Video: Numberphile, “The Mandelbrot Set,” with Holly Krieger
Who drew it first
A 1978 sketch, a 1980 IBM printout, a 1985 name
The set was not found by the man it is named for. Robert W. Brooks and Peter Matelski defined and drew it in 1978 while studying Kleinian groups, as a coarse character-printer image nobody paid much attention to. Benoit Mandelbrot produced the first proper visualization on 1 March 1980 at IBM’s Thomas J. Watson Research Center, with the computing power to see the filaments that the 1978 drawing could only hint at.
The name came five years later. Adrien Douady and John H. Hubbard established many of its fundamental properties in 1985 and named it in Mandelbrot’s honour. Their main result was that the set is connected: every black speck in the explorer, however isolated it looks at high zoom, is joined to the main body by a filament too thin to render. The same year, the August issue of Scientific American put the set on its cover and introduced the algorithm for computing it to a general audience, with images by Peitgen, Richter and Saupe at the University of Bremen. Drawing it has been a rite of passage for programmers ever since.
What the explorer is computing
The escape-time algorithm, pixel by pixel
Every pixel on the canvas is a value of c. The explorer iterates z² + c for that c and counts how many steps it takes for |z| to pass 2. Points that never escape within the iteration limit are painted black; the rest are coloured by their escape count, which is why the bands around the set look like contour lines. Black is the conventional colour for values that fail to escape before the limit, and the colour palettes in the “Aesthetic shift” panel are nothing more than different functions of that count.
Zooming raises the stakes. Near the boundary, neighbouring pixels can need hundreds of iterations to separate, so the explorer’s cost rises exactly where the picture gets interesting. The zoom multiplier shown at the bottom of the screen tells you how far you have gone; keep going and the double-precision arithmetic that browsers use eventually runs out of digits, at which point the image dissolves into blocks. That is a limit of the number format, not the set.
Reading the shape
The cardioid, the bulbs and the seahorses
The big heart-shaped body is the main cardioid, the region of c-values where the iteration settles to a single attracting fixed point. The circle to its left, radius 1/4 and centred on −1, is the period-2 bulb: c-values whose iteration ends up alternating between two values. Smaller bulbs around the cardioid correspond to higher periods.
The explorer’s co-ordinates menu jumps to neighbourhoods that fractal enthusiasts named long ago for what the filaments resemble once magnified: Seahorse Valley and Elephant Valley. “The Satellite” lands on a miniature copy of the whole set, one of countless copies embedded along the filaments. Each copy is slightly distorted and each carries its own smaller copies.
The boundary is as rough as a boundary can be
Dimension 2, and a conjecture still open
A smooth curve has dimension 1; a filled region of the plane has dimension 2. Mitsuhiro Shishikura proved that the Hausdorff dimension of the Mandelbrot set’s boundary equals 2: the edge of this set is as crinkled as a curve can possibly be while still being a curve. Whether that boundary also has positive area, meaning it takes up a measurable slice of the plane, is an unsolved problem.
The bigger open question is local connectivity. It is conjectured that the set is locally connected, which would mean that zooming in on any boundary point eventually shows a single tidy piece rather than infinitely many strands tangled together. Known as MLC, the conjecture has been proven for large classes of points but the full statement is still open. The explorer cannot prove anything, but it can show why the question is hard: pick a point on the boundary and keep zooming, and the strands keep arriving.
Every point is a map of another fractal
Julia sets, and what the Mandelbrot set indexes
Each value of c has a Julia set of its own: the set of starting points z (rather than parameters c) whose orbits under z² + c stay bounded. Some Julia sets are single connected blobs; others shatter into dust. The Mandelbrot set is the catalogue of which is which. A value of c belongs to the Mandelbrot set if and only if the corresponding Julia set is connected, which is why mathematicians describe it as a map of the connected Julia sets, or the connectedness locus of the quadratic family. The explorer draws the map; every pixel on it stands for a whole second fractal.
The bulbs carry arithmetic as well. For every rational number p/q, a hyperbolic component of period q bifurcates from the main cardioid at a point on its edge, so the bulbs tangent to the cardioid are indexed by fractions. Read the periods of the bulbs at successive scales between two large neighbours and the denominators follow the Fibonacci sequence. None of that was designed in. It falls out of z² + c.
The man behind the name
IBM, a coastline, and tenure at 75
Benoit Mandelbrot was born in Warsaw on 20 November 1924 and spent most of his working life outside the universities that would later claim him. He joined IBM in 1958 and stayed 35 years, becoming an IBM Fellow, with the computing access that made the 1980 image possible. His 1967 paper in Science, “How Long Is the Coast of Britain?”, made the case that the answer depends on the length of the ruler. He coined “fractal” in 1975 from the Latin fractus, broken, and published The Fractal Geometry of Nature in 1982.
Yale made him Sterling Professor of Mathematical Sciences and gave him his first tenured post in 1999, at 75, the oldest professor in the university’s history to receive it. He died on 14 October 2010, aged 85. His one-line manifesto survives him: “Clouds are not spheres, mountains are not cones, coastlines are not circles.”
How to use the explorer
Click to zoom, drag to pan, let the autopilot fly
Left-click zooms toward the cursor; drag the canvas to pan. The menu on the top left opens the flight controls: an autopilot that zooms continuously at a warp factor you set, zoom in and out buttons, the colour palettes, the co-ordinate presets and the current zoom multiplier. The X and Y readout in the top bar gives the complex co-ordinate at the centre of the view, so a location can be written down and found again.
Three things worth doing:
- Jump to The Satellite and keep zooming into its edge. The copy of the whole set is itself ringed with copies.
- Switch palettes while sitting in Seahorse Valley. Same escape counts, different colour functions, and the structure reads differently each time.
- Run the autopilot at the slowest warp and watch the filaments resolve. Structure appears in stages, and the stages never stop.
The explorer is a single page hosted at apps.x9m8.com/mandalbrot, with the source on GitHub. It runs entirely in the browser.
Further reading
Benoit Mandelbrot’s own account of why nature prefers rough shapes to smooth ones remains the place to start. The Fractal Geometry of Nature is the 1982 book in which Mandelbrot argues that coastlines, clouds and turbulence are better described by fractal dimension than by Euclid’s shapes, and it lays out the way of thinking that led to the set carrying his name. See it on Amazon (affiliate link).
The Fractal Geometry of Nature by Benoit B. Mandelbrot. The 1982 book that made the case for roughness as a measurable property of nature. See it on Amazon (affiliate link)
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For the cellular-automata side of the same family of ideas, simple rules producing unbounded structure, x9m8 has a companion simulator in Cellular Automata Explained: Game of Life, Rule 30 and Rule 110.
Sources
- Definition, history, Douady and Hubbard, Shishikura, the 1985 Scientific American cover, the cardioid and bulbs, and the MLC conjecture: Wikipedia, Mandelbrot set
- Mandelbrot’s life, the coining of “fractal”, IBM and Yale: Wikipedia, Benoit Mandelbrot
- Video: Numberphile, The Mandelbrot Set
- Book: The Fractal Geometry of Nature on Amazon (affiliate link)
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