Mandelbrot Set Explorer

Take a complex number c. Start at zero, square, add c, and do it again with the result. For c = 1 the values run 0, 1, 2, 5, 26 and are gone. For c = -1 they run 0, -1, 0, -1 for ever. The Mandelbrot set is the collection of every c of the second kind: the constants for which the orbit of zero stays bounded.

That is the whole definition, and it fits in a sentence. What does not fit anywhere is the result. The set has a boundary so intricate that no finite computation can draw it, and every image of it, from the first to the newest, is a statement about how far somebody was willing or able to compute. The honest way to look at a Mandelbrot picture is as a measurement with an error bar.

A map printed in asterisks

The first published picture is not a glowing spiral. It is a block of asterisks on page 71 of a conference volume, “Riemann Surfaces and Related Topics,” published by Princeton University Press in 1980 from a 1978 meeting at Stony Brook. Robert Brooks and J. Peter Matelski were studying something else, discrete groups of matrices, and needed to know for which constants the map had a stable periodic orbit. Their printout shows a fat heart shape with a round lobe on its left and a line running out to the point marked (-2, 0), with (1/4, 0) marked at the other end.

Benoit Mandelbrot came to the same object from the pictures. By his own account he found the set in 1980 during a stay at Harvard, on a newly installed Vax computer in a basement, working at night at a faint screen while a teaching assistant, Peter Moldave, wrote the programs. Their first rough image is dated 1 March 1980. Better pictures followed on equipment at IBM, where Mandelbrot worked, and his article appeared the same year in the Annals of the New York Academy of Sciences.

Others were drawing too. John Hubbard has described a minicomputer at Orsay that spent the spring of 1977 printing pictures of a related problem in X’s, 0’s and 1’s. He went on to plot Julia sets, and it seems he showed his pictures to Mandelbrot that year. So the story has several beginnings: a character printer in 1977, the asterisks of 1978 and a screen in 1980.

The name came from the people who did the hard mathematics. Adrien Douady and John Hubbard obtained their main results in 1981 and 1982 and wrote them up as course notes at Orsay in 1983 and 1984. They named the set after Mandelbrot, and they proved the fact that pictures of the time could not show: the set is connected. The specks that seem to float free around it are joined to the main body by filaments too thin for a coarse grid to catch.

The rule that lets a computer stop

A program cannot follow an orbit for ever, so it needs a test that settles the matter early. There is one, and its proof is four lines long. If an iterate z is farther than 2 from zero, and at least as far out as c is, then the next iterate is larger again by a factor of more than |z| - 1, which exceeds 1. The sizes grow geometrically and never come back. So the moment an orbit leaves the circle of radius 2, c is outside the set, and the program can stop and record how many steps it took.

That step count is what the colors mean. A picture of the Mandelbrot set is almost entirely a picture of its outside: each exterior point is tinted by how quickly its orbit escaped, and the set itself is usually one flat color in the middle. The bright bands that crowd toward the edge are places where escape takes longer and longer.

Counting in whole steps makes stripes. They can be removed with a correction that uses how far past the circle the orbit landed: the smooth count is n + 1 minus the base-2 logarithm of the natural logarithm of the size of the escaping iterate, where n is the step on which it escaped. A worked case, with a generous escape radius of 256: for c = 1 the orbit continues 26, 677, so n = 5 and the size is 677. The natural logarithm of 677 is 6.5177, its base-2 logarithm is 2.7044, and the smooth count is 6 - 2.7044 = 3.2956. This tool computes exactly that number for that point.

A second quantity comes almost for free. Carry the derivative of the orbit with respect to c alongside the orbit, and at escape the ratio 2 |z| ln |z| / |z’| approximates a bound b on the distance from c to the set: no point of the set lies nearer than a quarter of b. Renderers use it to shade the neighborhood of the boundary, which brings out filaments far narrower than a pixel.

The other side of the test is harder. An orbit that has not escaped after a thousand steps may escape on the next one. Every renderer therefore has an iteration limit, and every point that reaches it is drawn as inside. Near the cusp at c = 1/4 this is easy to see: a point a hundred-millionth to the right of the cusp takes 31,417 steps to get out by this tool’s count. Set the limit too low and the set appears to swell.

Why the edge never smooths out

Zoom into a circle and it flattens into a line. Zoom into the edge of the Mandelbrot set and it does not flatten at all. Mitsuhiro Shishikura proved in 1991 that the boundary has Hausdorff dimension 2, the same dimension as a filled region of the plane, although it is a curve-like object with no inside of its own. It is, in a precise sense, as crinkled as a boundary in a plane can be.

Part of what turns up on the way down is the set itself. Small copies sit along every filament, each with its own heart and lobe. The lobes carry meaning too: for every c inside the big one on the left the orbit of zero settles into a cycle of two points, and each bulb attached to the main body has its own cycle length, which is also the number of spokes on the antenna it wears.

Then there are the Julia sets. Fix c and ask instead which starting points stay bounded; the answer is the filled Julia set for that c. A theorem of Fatou and Julia from 1919, the one Douady and Hubbard start their notes with, says it comes in exactly two kinds. If zero stays bounded, the Julia set is one connected piece. If zero escapes, it is a dust of disconnected points, sometimes called Fatou dust. The Mandelbrot set is therefore the index to all of them: stand on a point of the map and the Julia set you get is whole or shattered according to which side of the boundary you are on.

Moving the pointer across an edge of the Mandelbrot set while watching the Julia set fall apart is the fastest way to believe the theorem.

Where ordinary arithmetic gives out

The numbers in almost every program are IEEE 754 doubles, which carry 53 bits of precision, about sixteen decimal digits. A view of the whole set a thousand pixels wide needs pixels about 0.004 apart. Magnify ten trillion times and neighboring pixels differ only in the sixteenth digit. A little further and they are the same number, and the picture dissolves into blocks.

The direct cure is to compute every pixel with hundreds or thousands of bits, which is correct and painfully slow. The cure used now is a trick of bookkeeping that K. I. Martin made popular in 2013. Compute one orbit, that of the center of the view, in high precision. For every other pixel track only its difference from that orbit, which obeys a recurrence of its own and is small enough to live comfortably in a double. One slow orbit, a million fast ones.

It fails in a specific way. When the true orbit of a pixel passes close to zero, the difference is nearly as large as the reference and the low digits that tell neighboring pixels apart are lost; patches of the image go flat or noisy. These are called glitches. A forum contributor known as Pauldelbrot found a test that detects them, and a later idea credited to Zhuoran avoids them: when the pixel’s orbit is nearer to zero than its difference is, restart the comparison from the beginning of the reference orbit.

The people who build these renderers say plainly that no complete proof of correctness exists for the method. That is a reason to test rather than to trust. This tool compares its fast pixels against slow exact ones at depths down to its limit, a magnification of 1.5e250, and stops there with a message instead of guessing. One number a reader can check: pick the place called “Small copy by the spiral, 7.5e51 x” and the automatic iteration limit reads 16,799, which is 256 + 96 times the base-2 logarithm of the magnification.

Even exact arithmetic for the center does not make every pixel exact. A sample that lies within a billionth of a pixel of the set can still get a count that is off, because the count changes faster there than a 53-bit difference can follow. Such samples are rare and they sit on the boundary, where the picture is a blur of bands in any case.

What nobody knows

The exact area of the set remains unknown. Counting pixels at a resolution of almost 88 trillion gave an estimate of 1.50659, and the OEIS prints 1.50659177 with an uncertainty of 8 in the eighth decimal place. A series formula exists as well, and five million of its terms give a proved upper bound of 1.68288. The gap between the two figures has not been explained: either the series converges very slowly or the boundary itself takes up area.

The central open question is older than most of the pictures. Douady and Hubbard conjectured that the set is locally connected, a property abbreviated MLC, and wrote that they did not know how to prove it. It would imply that every region inside the set is one where orbits settle onto an attracting cycle. Jean-Christophe Yoccoz proved it at a large class of points; a 2023 paper by Dzmitry Dudko and Mikhail Lyubich settled it at the classical period-doubling point, where it had still been open. The general statement remains a conjecture.

So the object defined in one sentence has been drawn for more than forty years, proved connected, measured to dimension 2 along its edge, and zoomed past any physical scale, and its basic topology is still not settled. Each picture shows the part someone computed. The set is what is left when the computing never stops.