The Mandelbrot Set, named after one Benoit Mandelbrot, who spent years studying and popularizing it. We at psnt.net are grateful for Wolfgang Beyer, who created these beautiful images and freely released them for the world to enjoy. Source of all images: Wikimedia Commons
It’s Art Sunday again! This week we bring to you the Mandelbrot Set. This is a mathematical object; that is, it is a particular set of points. It is produced by inputting a number into a simple formula (and we do mean simple; it’s 8th-grade stuff, nothing more complicated than a square: z^2 ) and getting an output. Then one takes that output and puts it back into the same formula. So you do this over and and over again (this process of feedback is called iteration). Then, depending on what happens to your output over many loops, you either plot a point or you don’t. Repeat for another point, and then again, etc.
The clip-and-save messages are three: (1) the math is simple and can be done by anyone (and we mean anyone) on a super-cheapo calculator after about 10 minutes of instruction, and (2) the only hard part is that it has to be done over and over and over and over and over and over and over (that’s why things like the Mandelbrot Set were never really discovered until after the advent of the computer); and (3) the result is a mathematical object that is insanely beautiful and infinitely rich. There is no bottom to it; our viewing of it is limited only by our computing power and our patience. And the complexity increases the deeper one drills.
The philosophical cash-out is: The creation of complex objects does not require complex rules.
The philosophical question is: Does this thing really exist? It’s basically the same as asking, does a circle exist? How about the idea of a circle? Here we’re drifting into Platonism and notions of the reality of ideas. Put it this way: Is mathematics discovered or is it invented? We at psnt.net are definitely fans of Plato and we say: discovered. But ours (and Plato’s) is not the only opinion.
The religious question is: Are there religious overtones to mathematics, or is that just a bunch of hooey?
For those who would like to further explore the Mandelbrot Set and other cool fractal objects, here are some places to check out. You can zoom in on any point of the set you wish at this site. Also, this animated gif takes you through the sequence below, but keeps on going and going. Not to be outdone, a group calling themselves “teamfresh” has created a 14-minute drilldown that has an incredible first-frame-to-last-frame ratio of 2760:1. It’s a great video, but unless you love technopop you may want to turn down the volume before you watch it. You may also be interested to learn about Julia sets, which are close cousins of the Mandelbrot Set and which I love, not least because they carry the name of my eldest daughter.
UPDATE 3/21 Thanks go out to Alert Reader Brice Harris, who pointed us toward this wonderful NOVA video that explores the connections between fractals and the natural world. On this page you can also find links to an interview with Benoit Mandelbrot, another fun interactive Mandelbrot Set, and you can even design your own fractal. Isn’t the Web wonderful?
*For those who’d like to know the mathematical details, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial zn+1 = zn2 + c remains bounded. That is, a complex number, c, is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn never exceeds a certain number (that number depends on c) however large n gets. Definition due to Wolfgang Beyer via Wikipedia
AND NOW, DOWN THE RABBIT HOLE
The sequence below was created by Wolfgang Beyer, who writes, “The magnification of the last image relative to the first one is about 10,000,000,000 to 1. The final image, shown on an ordinary monitor, corresponds to a full Mandelbrot set with a width of 4 million kilometers (10 times greater than the earth-moon distance!). Its border would show an astronomical number of different fractal structures.”
Each image is contained by the white zoom box in the image above it. Click on any picture for a beautiful high-resolution (several MB) version. The different colors are related to how fast the output value of the formula changes relative to its input value. Take your time, stop to smell the fractal flowers, and soak up the wonder of it all. One thing I really love is the way the black bug-shaped motif shows up on every scale, but it’s never exactly the same twice.