# Let's get stoned and explore bounded equations

May 24, 2005 5:16 PM Subscribe

Born in Poland, grew up and got PhD in France, employed at Yale and IBM in New York.

posted by gleuschk at 5:36 PM on May 24, 2005

posted by gleuschk at 5:36 PM on May 24, 2005

Ah, there goes the rest of the afternoon.

posted by Joey Michaels at 5:44 PM on May 24, 2005

posted by Joey Michaels at 5:44 PM on May 24, 2005

Sorry about the provenance of Benoit M., Should have researched. Also, cut + paste the second url dunno why link failed.

posted by longsleeves at 6:02 PM on May 24, 2005

posted by longsleeves at 6:02 PM on May 24, 2005

These will go nicely on my Trapper Keeper

posted by MiltonRandKalman at 6:23 PM on May 24, 2005

posted by MiltonRandKalman at 6:23 PM on May 24, 2005

I suppose the significance of this mathmatical shape couldn't be explained to the casual mathematician?

posted by Citizen Premier at 6:39 PM on May 24, 2005

posted by Citizen Premier at 6:39 PM on May 24, 2005

The Mandelbrot set, named after Benoit Mandelbrot, is a fractal. Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to the large-scale characteristics. Although the Mandelbrot set is self-similar at magnified scales, the small scale details are not identical to the whole. In fact, the Mandelbrot set is infinitely complex. Yet the process of generating it is based on an extremely simple equation involving complex numbers.

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Understanding complex numbers

The Mandelbrot set is a mathematical set, a collection of numbers. These numbers are different than the real numbers that you use in everyday life. They are complex numbers. Complex numbers have a real part plus an imaginary part. The real part is an ordinary number, for example, -2. The imaginary part is a real number times a special number called i, for example, 3i. An example of a complex number would be -2 + 3i.

The number i was invented because no real number can be squared (multiplied by itself) and result in a negative number. This means that you can not take the square root of a negative number and get a real number. When you take the square root of a number, you find a number that can be squared to get that number. The number i is defined to be the square root of -1. This means that i squared is equal to -1. So when you square an imaginary number you can get a negative number. For example, 3i squared is -9.

Real numbers can be represented on a one dimensional line called the real number line. Negative numbers like -2 are plotted to the left of zero and positive numbers like 2 are plotted to the right of zero. Any real number can be graphed on the real number line.

Since complex numbers have two parts, a real one and an imaginary one, we need a second dimension to graph them. We simply add a vertical dimension to the real number line for the imaginary part. Since our graph is now two-dimensional, it is a plane, the complex number plane. We can graph any complex number on this plane. The colored dots on this graph represent the complex numbers [2 + 1i], [-1.5 + 0.5i], [2 - 2i], [-0.5 - 0.5i], [0 + 1i], and [2 + 0i].

Graphing the Mandelbrot set

The Mandelbrot set is a set of complex numbers, so we graph it on the complex number plane. However, first we have to find many numbers that are part of the set. To do this we need a test that will determine if a given number is inside the set or outside the set. The test is based on the equation Z = Z2 + C. C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing, the point on the complex plane that will be plotted when testing is complete. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.

We're not really interested in the actual value of Z as it changes, we just look at its magnitude. The magnitude of a number is its distance from zero. For example, the number -9 is a distance of 9 from zero, so it has a magnitude of 9. The magnitude of a complex number is harder to measure. To calculate it, we add the square of the number's distance from the x-axis (the horizontal real axis) to the square of the number's distance from the y-axis (the imaginary vertical axis) and take the square root of the result. In this illustration, a is the distance from the y-axis, b is the distance from the x-axis, and d is the magnitude, the distance from zero.

As we iterate our equation, Z changes and the magnitude of Z also changes. The magnitude of Z will do one of two things. It will either stay equal to or below 2 forever, or it will eventually surpass two. Once the magnitude of Z surpasses 2, it will increase forever. In the first case, where the magnitude of Z stays small, the number we are testing is part of the Mandelbrot set. If the magnitude of Z eventually surpasses 2, the number is not part of the Mandelbrot set.

posted by longsleeves at 6:47 PM on May 24, 2005

--------------------------------------------------------------------------------

Understanding complex numbers

The Mandelbrot set is a mathematical set, a collection of numbers. These numbers are different than the real numbers that you use in everyday life. They are complex numbers. Complex numbers have a real part plus an imaginary part. The real part is an ordinary number, for example, -2. The imaginary part is a real number times a special number called i, for example, 3i. An example of a complex number would be -2 + 3i.

The number i was invented because no real number can be squared (multiplied by itself) and result in a negative number. This means that you can not take the square root of a negative number and get a real number. When you take the square root of a number, you find a number that can be squared to get that number. The number i is defined to be the square root of -1. This means that i squared is equal to -1. So when you square an imaginary number you can get a negative number. For example, 3i squared is -9.

Real numbers can be represented on a one dimensional line called the real number line. Negative numbers like -2 are plotted to the left of zero and positive numbers like 2 are plotted to the right of zero. Any real number can be graphed on the real number line.

Since complex numbers have two parts, a real one and an imaginary one, we need a second dimension to graph them. We simply add a vertical dimension to the real number line for the imaginary part. Since our graph is now two-dimensional, it is a plane, the complex number plane. We can graph any complex number on this plane. The colored dots on this graph represent the complex numbers [2 + 1i], [-1.5 + 0.5i], [2 - 2i], [-0.5 - 0.5i], [0 + 1i], and [2 + 0i].

Graphing the Mandelbrot set

The Mandelbrot set is a set of complex numbers, so we graph it on the complex number plane. However, first we have to find many numbers that are part of the set. To do this we need a test that will determine if a given number is inside the set or outside the set. The test is based on the equation Z = Z2 + C. C represents a constant number, meaning that it does not change during the testing process. C is the number we are testing, the point on the complex plane that will be plotted when testing is complete. Z starts out as zero, but it changes as we repeatedly iterate this equation. With each iteration we create a new Z that is equal to the old Z squared plus the constant C. So the number Z keeps changing throughout the test.

We're not really interested in the actual value of Z as it changes, we just look at its magnitude. The magnitude of a number is its distance from zero. For example, the number -9 is a distance of 9 from zero, so it has a magnitude of 9. The magnitude of a complex number is harder to measure. To calculate it, we add the square of the number's distance from the x-axis (the horizontal real axis) to the square of the number's distance from the y-axis (the imaginary vertical axis) and take the square root of the result. In this illustration, a is the distance from the y-axis, b is the distance from the x-axis, and d is the magnitude, the distance from zero.

As we iterate our equation, Z changes and the magnitude of Z also changes. The magnitude of Z will do one of two things. It will either stay equal to or below 2 forever, or it will eventually surpass two. Once the magnitude of Z surpasses 2, it will increase forever. In the first case, where the magnitude of Z stays small, the number we are testing is part of the Mandelbrot set. If the magnitude of Z eventually surpasses 2, the number is not part of the Mandelbrot set.

posted by longsleeves at 6:47 PM on May 24, 2005

The most versatile fractal app is Ultrafractal. Check out prior fractal-related FPPs.

posted by Gyan at 7:00 PM on May 24, 2005

posted by Gyan at 7:00 PM on May 24, 2005

My favourite Mandelbrot viewer is XaoS. If that Mandelbrot Explorer is cool, Xaos is bind-bendingly awesome. If you have a fast computer. Set iterations to at least 1000, then zoom in with the mouse... It can do something like 15 real-time frames a second of the stuff I remember waiting twenty minutes for my IBM PC to draw.

posted by sfenders at 7:10 PM on May 24, 2005

posted by sfenders at 7:10 PM on May 24, 2005

i can never see a mandelbrot set now without thinking of the movie Pi... Fractals are damn interesting

posted by 0bvious at 7:19 PM on May 24, 2005

posted by 0bvious at 7:19 PM on May 24, 2005

Buddhabrot

Full disclosure: self-link buried somewhere in those results.

posted by polyglot at 7:51 PM on May 24, 2005

Full disclosure: self-link buried somewhere in those results.

posted by polyglot at 7:51 PM on May 24, 2005

I spent hours and hours and hours running Fractint when I was in college. It's in version 20.0 now!

posted by Songdog at 7:58 PM on May 24, 2005

posted by Songdog at 7:58 PM on May 24, 2005

Zoom in really deep on the "cleft" at the "rear" of the curve.

It's like a cross-sectional side view of goatse. Hairy and everything.

posted by Eideteker at 8:02 PM on May 24, 2005

It's like a cross-sectional side view of goatse. Hairy and everything.

posted by Eideteker at 8:02 PM on May 24, 2005

Yes sfenders!

It'd surprise 15-year-old me to think that the mandelbrot set would seem retro someday, but yeah...

People of 1994! Real-time, zooming XaoS welcomes you to 1999!

posted by tss at 8:07 PM on May 24, 2005

It'd surprise 15-year-old me to think that the mandelbrot set would seem retro someday, but yeah...

People of 1994! Real-time, zooming XaoS welcomes you to 1999!

posted by tss at 8:07 PM on May 24, 2005

>People of 1994! Real-time, zooming XaoS welcomes you to 1999!

Thanks. It's nice here, but coffee costs too much. People of 1999, work for change!

posted by longsleeves at 9:16 PM on May 24, 2005

Thanks. It's nice here, but coffee costs too much. People of 1999, work for change!

posted by longsleeves at 9:16 PM on May 24, 2005

Hey, I used this for a presentation last December!

Keeping with the line of sarcrassisity, aren't all the other sciences just secondary biatches of mathematics? :)

posted by buzzman at 10:57 PM on May 24, 2005

Keeping with the line of sarcrassisity, aren't all the other sciences just secondary biatches of mathematics? :)

posted by buzzman at 10:57 PM on May 24, 2005

Apparently Gaston Julia deserves most of the credit for first studying these kinds of sets. It's odd too, because he has no nose. The deal with the m-set is basically this: if you take a number, and square it repeatedly it will either get sucked down to zero or shoot off to infinity (or hang out on the boundary |z| = 1). This corresponds to iterating the equation z |-> z^2. Now pick a number c and do a similar process, except pick for once and for all a seed value, square it and add c, square that and add c, etc. This corresponds to iterating the equation z |-> z^2 + c. If you draw a map for each number c of whether z shoots off to infinity or not (and color by how fast), that's the mandelbrot set. The fact that the equation z^2 + c is so ridiculously simple and the resulting shape so insanely complex and beautiful, is something of religious significance to me. By the way if complex numbers bother you, they shouldn't. Despite their strange and mysterious behavior they are in many ways more "natural" than the reals, being the algebraic closure thereof.

posted by Astragalus at 4:05 AM on May 25, 2005

posted by Astragalus at 4:05 AM on May 25, 2005

I've got to admit that I find the math that lies behind these things more beautiful that the diagrams, which I've always found pretty ugly. Sorry.

posted by blindsam at 4:42 AM on May 25, 2005

posted by blindsam at 4:42 AM on May 25, 2005

There is a showcase worth checking out from Gyan's Ultrafractal link.

Here are some more fractal images that help give you an idea of their infinite scope. Tell me those aren't awesome, blindsam.

posted by foraneagle2 at 3:13 AM on May 26, 2005

Here are some more fractal images that help give you an idea of their infinite scope. Tell me those aren't awesome, blindsam.

posted by foraneagle2 at 3:13 AM on May 26, 2005

Wow, after installing XaoS, I don't recommend the above links anymore. I recommend XaoS!

posted by foraneagle2 at 3:22 AM on May 26, 2005

posted by foraneagle2 at 3:22 AM on May 26, 2005

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posted by AlexReynolds at 5:23 PM on May 24, 2005