Hunting the Hidden Dimension

A Sense of Scale

So, naturalists observe, a flea
Has smaller fleas that on him prey;
And these have smaller still to bite 'em,
And so proceed ad infinitum.
—Jonathan Swift, from "On Poetry: A Rhapsody"

The satirist and author of Gulliver's Travels might have been talking about fractals when he made this oft-quoted observation—if he hadn't lived two centuries before fractals were discovered. (In fact, Swift was complaining about lesser poets criticizing better ones, like pestering fleas.) As it happens, these four lines can serve as a perfect metaphor for the infinitely detailed, "self-similar" nature of fractals. In this interactive, zoom deep into a Mandelbrot set, the most famous of fractals, to the mind-bending magnification of 250,000,000x. Along the way, you'll see what is meant by self-similarity, how the iconic Mandelbrot-set shape keeps turning up at smaller scales like one of Swift's ever-tinier fleas, and why the 18th-century wit's metaphor suits fractals to a tee.—Peter Tyson

Note: NOVA Online's Rachel VanCott used the program Ultra Fractal 5 to generate these images of the Mandelbrot set.

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The black, flea-like shape you see here is the classic Mandelbrot-set icon. Okay, it looks more like Frosty the Snowman tipped over into an inkwell. But in honor of Jonathan Swift (see intro), suspend disbelief for a moment and imagine it as a kind of fractal flea, seen from above, with its bristly head facing left. Now, see the next image to zoom in on our fractal flea's head.

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Having magnified the image 10 times, the first thing you might notice, beyond the great "V" of blue, is that the icon—our fractal flea—reappears, in smaller form, all along the edge of the flea. Our flea is positively infested with Mandelbrot-set icons, which get tinier and tinier as your eye drifts down into the crease. Swift would like where this is going, don't you think?

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Now you can see those tiny fractal fleas more closely. They're not exact replicas of the Big Flea at the start, but they're pretty close. In fact, this fractal, like all fractals, is "self-similar." No matter how much you magnify certain portions of the colorful boundary area of a Mandelbrot set, you will come upon particular shapes, like our fractal flea, that closely resemble shapes you've already seen at lower magnification.

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Wow, rather frondy, if that's a word. Where did all the fractal fleas go? Well, a Mandelbrot set includes more than just the classic icon we started with. In fact, any portion of the images you see in this feature that is black belongs to the Mandelbrot set that was generated by the iterative equation we used. (For more on iterative equations and how they generate fractals, see The Most Famous Fractal.) But wait... What is that hidden amidst this filamentous frond? You got it—another Mandelbrot-set icon.

Primula Wilsonii
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See how similar it is to our Big Flea? It just looks a little frothier around the edges, perhaps. Amazingly, the simple equation that furnished our original Mandelbrot-set icon created this one, as well as all the swirly filaments and colors at this and all other powers of magnification. Who knew math could be so artistic?

Regal Lily
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We've now magnified our view 20,000 times, and the big round shape to the right is the head of our previous fractal flea. Like the 60-foot-tall farmer who finds Gulliver when he's marooned in the Land of Brobdingnag—we can't forget Swift, can we?—the larger flea dwarfs yet another, even tinier flea clinging to its long "mouthparts." (Run with us here a bit.)

Paperback Maple
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The edges of this fractal flea are a little less well-defined than those of earlier ones, but it is still easily recognizable as a Mandelbrot-set icon. So how can we keep zooming in like this and still find such vivid detail? Well, each of these images represents a graph of the complex number plane, and between any two numbers on the plane exists an infinite number of numbers. So a Mandelbrot set, by definition, is infinitely detailed.

Peach Tree
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We are now half a million times smaller than when we started. And Swift's ever-smaller fleas? Still here, pressed up against this particular flea like—well, like Lilliputians tossing ropes over the unconscious Gulliver.

Peach Tree
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Is your head hurting yet? Five million times smaller. Our image is beginning to look like a leaf with its delicate tracery of veins. That's fitting, because a tree is a good example of the fractal nature of Nature. Branches branch into smaller branches, which themselves branch into yet smaller branches, all roughly similar to any other branch on the tree. Same goes for the veins in a leaf.

Peach Tree
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Even more like we're inside a leaf, veins shooting out everywhere. And, of course, the ubiquitous icon is here. (It may look more like a mini space shuttle at this point, but it's still a Mandelbrot-set icon.) Though you can't see 'em, the icon has its own "smaller fleas that on him prey," of course. "Thus every poet, in his kind/Is bit by him that comes behind," writes Swift in the lines that follow his flea metaphor, "Who, though too little to be seen/Can teaze, and gall, and give the spleen..."

Peach Tree
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We could proceed ad infinitum, magnifying ever further, but we don't want to give you the spleen. So just how tiny have we gotten, at a magnification of 250,000,000x? Well, if our Big Flea was an actual flea, you are now looking at something smaller than an atom of carbon within that flea. Or, if what appears here is our actual flea, then our Big Flea is about the size of Switzerland. That's one Brobdingnagian flea, eh, Mr. Swift?


We recommend you visit the interactive version. The text to the left is provided for printing purposes.

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© | Created October 2008