Handprint : Mandelbrot

The Mandelbrot Set in Powers of 10	The Mandelbrot set results from the output of a dynamical process and a set of mapping or coloring rules. Even so, it's easy to think of it as if it were an object rather than a process. Some books on chaos talk about "places" on the Mandelbrot set, and delve its microscopic landscape in a picturesque way. Or they portray it as a craggy mountain landscape or as an emblem on the surface of an imaginary planet. These renderings emphasize the beauty and stupefying complexity of the Mandelbrot set, but obscure the fact that it represents the specific outcomes of a nonlinear process, a Verhulst equation. One of my favorite science books, Powers of Ten, traverses the known physical world in successive views one-tenth the scale of the last. Here I've taken the same approach to the Mandelbrot set to contrast it with the objects of the physical world. Unlike physical objects, which eventually dissolve into discrete atoms and quarks, the structure of a nonlinear process is often entirely independent of scale. We dive far into the fractal intricacies of the Mandelbrot set only to discover replicas of the original landscape. This highlights the difference between indeterminacy and chance: chaotic systems are not random. Computed nonlinear processes like the ones that create the Mandelbrot set are, for all their complexity, exactly determined. Start with the same values (in the Mandelbrot set, the values of a and ib), the same computer hardware, and the same mapping rules, and you will always get exactly the same fractal image. Indeterminacy arises because this complexity entirely depends on the exact initial state and physical form of the dynamical process - and these are often beyond our ability to forecast or control. The only way we can resolve the indeterminacy is to run the process and see what happens. This informational asymmetry -- indeterminate processes emerging from imperceptible origins -- gives dynamical systems their two signature properties. Their "randomness" arises because they massively amplify unknowably small variations in atomic information or numerical storage or physical processes, which can vary beyond our measurement capabilities. But the indeterminacy arises because the only way the consequences of those variations can be expressed is through the process itself -- and therefore through local events in the real world. The Mandelbrot set is not an object, yet it is inherently defined by a process located in the real world. It is an image of the origins of the world at the limits of information, a contemplation of the dynamics that can amplify atomic details into global events. Stepping through the set in powers of 10 hints at the range of scale this amplification can span. START


NEXT	Base image. The entire Mandelbrot set, with the dimension of real numbers as the x axis. All parts of the Mandelbrot set are within the frame, and therefore all parts of the prisoner set created by the Verhulst process. At the top and bottom are two large "buds" or circular areas extending off the main body of the set, similar to the large bud attached at the left. Extending from each is a Y-shaped branch of color, located analogously to the straight "antenna" of color extending to the left of the lefthand bud. This is the feature we will pursue through all levels of detail. The white rectangle indicates the area to be enlarged: it is 1/10 the dimensions (1/100 the area) of the starting image.

NEXT	Power $10$ ^-1. The Y-shaped branch of color takes up most of the image, with the top of the bud at the bottom of the screen. The branch turns out to contain numerous smaller replicas of the Mandelbrot set, all of different sizes, of which the largest appear as tiny black dots. The largest of these replicas is about halfway along the lefthand (longer) branch. This gives us a simple descent algorithm: begin with a Mandelbrot set find the large bud on the righthand (top) side of this set find the Y-shaped branch coming out of the bud find the largest Mandelbrot set in the lefthand arm of this branch. Repeat indefinitely.

NEXT	Power $10$ ^-2. This image shows that the replica Mandelbrot set is almost identical to the main set. But not exactly: there are more and longer branches of color extending off the buds on the outside circumference of the set. We again see two large buds on either side of the main body, and the Y-shaped branches of color extending from them.

NEXT	Power $10$ ^-3. The branch again has a number of Mandelbrot copies embedded in it, with the largest replica of the set about halfway along the lefthand branch. We are moving through levels of scale by following the "self-similarity" of a fractal, which means that characteristic details will reappear across the periodically smaller levels of scale.

NEXT	Power $10$ ^-4. A second difference between the replica and main Mandelbrot sets appears in the asymmetry between the lefthand and righthand sides of the replica. This is easiest to see in the pattern of the branches of color extending off either side, although you'll also see that the lefthand side of the replica is slightly larger than the right. These tiny variations distinguish one replica of the Mandelbrot set from all others. Each is unique. Thus, the self-similarity of the Mandelbrot set is not exact. Exact self-similarity appears in a related family of fractals, called Julia sets, that also arise from the Verhulst process.

NEXT	Power $10$ ^-5. This image represents 1/100,000th of the first image. Taking the first image to represent the Mandelbrot set in actual size, the large bud in this picture, to scale, would be roughly 0.00005 centimeter across -- about the size of a cold virus. Yet branches and buds appear as detailed and perfectly formed as in the main Mandelbrot set. Not only does the Y-shaped branch contain uncountable copies of the Mandelbrot set, but the scalloping of small buds around the circumference of the large bud also emit branches from their sides and tips that contain vast numbers of copies of the Mandelbrot set. We can never be precise about how many replicas of the Mandelbrot set there are, because this depends on the level of detail we choose to count them on.

NEXT	Power $10$ ^-6. As we traverse smaller replicas of the Mandelbrot set, the branches of color seem to grow. We can now see a profusion of fainter tendrils that extend much farther from the body of the replica set. The entire frame is traced with these thin wisps of color, each marking differences in the outcome of the Verhulst process. This is a general feature of the Mandelbrot set. The main set is entirely surrounded by a minute dense web of such tendrils. Although the dark red branches in this image contain replicas of the Mandelbrot set, the fainter tendrils do not: they are part of the escape set. Even so, they do contain infinite variations of branching, scalloping or scrolling color that echo the chaotic orbits of the nearest points inside the Mandelbrot set.

NEXT	Power $10$ ^-7. Still another bud in a branch. The colors are more intense because the iterations required to resolve the set have increased substantially. We now must iterate some starting values of z up to 100,000 times before they jump into the escape set.

NEXT	Power $10$ ^-8. Another bud in the branch. Note the much smaller copies of the Mandelbrot set visible as yellow specks further along the branch of color to the right. All of the dark red tendrils spiking off the main branch contain more copies of the set, all with their own branches off buds. It is as if, by penetrating a hall of mirrors, more mirrored hallways appear. Further down in the details of the set the landscape seems to become more complex and intricate, not less so. Increasing computational power and accuracy does not simplify the process, but makes it more complicated and minutely contrasted. We realize that in a chaotic system measurement can never reach certain knowledge.

NEXT	Power $10$ ^-9. This frame gives a good sense for how far out from the replica set the branches of color extend. The small clumps of darker color in the branches mark junctures where replicas of the branching patterns (and perhaps replicas of the set) appear. In other parts of the Mandelbrot set the tendrils weave into turbulent but highly patterned branches and scallops of color. They take on recognizably different textures in different parts of the set. As we move along a descent algorithm that depends on self-similarity, we reveal patterns of local uniqueness.

NEXT	Power $10$ ^-10. At each reappearance, the replica set has rotated by about 35° from its original orientation and has reduced to about 0.67% its previous size. These constants indicate that one could predict the position of all further replicas of the set as far down into the fractal landscape as numerical accuracy allowed. This spiraling recurs throughout the set. In some places, the spirals close around a point, coiling inwards like the walls of an endless tunnel toward the infinitely small. A simple illustration that every part of the Mandelbrot set, across all levels of detail, is governed by a tightly organized and coherently integrated landscape of dynamical variation.

NEXT	Power $10$ ^-11. The ragged appearance of the bud at the top of the image indicates that we are nearing the limits of computational resources. The boundary between the escape and prisoner sets is dissolving into numerical inaccuracy created by limits on numerical precision or the number of iterations on my computer. This even though the image was created using as many as 1 million iterations per pixel with real numbers capable of representing over 100 decimal places.

TOP	Power $10$ ^-12. And here we stop: at the replica Mandelbrot set in the branch off the bud on the replica set in the branch off the bud on the replica set in the branch off the bud on the replica set in the branch off the bud on the replica set in the branch off the bud on the replica set in the branch off the bud on the main Mandelbrot set. This image is one trillionth the scale of the base image. If we take the base image to represent actual size, then the Mandelbrot set at the center of this frame, to scale, would be less than .0001 nanometer across - much smaller than any atomic particle, and about 1/35,000 the diameter of an atom of oxygen. Conversely, if we take this frame to represent actual size, then the Mandelbrot set in the base image would be, to scale, approximately 76 million kilometers long -- more than half the distance between the Earth and the sun. The tiny set in this image is degraded by computational limitations, yet it is clearly a replica of the original that more powerful computers and software could resolve to still deeper layers of detail. Yet even those tools would find still more complexity and indeterminacy beyond their reach. Even so, each pixel in this image represents a value for (a,b) that will drive the behavior of the Verhulst equation, as computed on my computer, in an entirely deterministic way. TOP
Credit to Adam Smith's Floating Fractals 5.1 for the figures shown here. This program is available for free at shareware.com and other shareware sites on the Web.