4/26/2004: Buddhabrot

[Undertoad]

Thanks to mrputter for pointing this out.

Melinda Green developed different ways to display the Mandelbrot Set and found that they became rather Buddha-esque. The above is possibly the least interesting of all of them, so go to the page and check them out.

What's the Mandelbrot Set, you ask. Here's a quick introduction to it. It's hard math... basically, it's a mathematical function that is a fractal. What's a fractal, you ask? You ask some damn difficult questions don't you? I didn't major in math!

Part of the point, I guess, is that certain things are defined as just a bunch of little things just like it. Which is something you might not be able to see right away, unless you look differently at it. Like one guy did with this cauliflower, to display its fractal qualities. In computer science we call that recursion, although on reading the Wikipedia entry on fractals I learned that they are not all recursive, and so I don't think I understand fractals exactly. (Paranoid, do you want to take a shot here?)

Note that even though the images resemble Hindu art, they were actually generated completely automatically, without any sort of human artistic intervention. When I first tried using the new technique, I had no idea what the images might look like and was completely surprised by the results.

Figure that this Buddha-like image is made up of infinite number of smaller Buddhas, and if you looked closer at it (especially by examining the graph produced by fractions with increasingly significant digits) you'd see them. And now, your mind should be completely blown, eh?

The spiritual amongst us might take some meaning in finding the Buddha in a mathematical formula. The skeptical amongst us well remember the "face on Mars", or the Virgin Mary in a tree stump, or the Virgin Mary in the side of a fence, and the countless children examining clouds for figures they could see. We find patterns; it's what we do, we humans.

And this is only a two-dimensional graph anyway. When they find a function to produce a three-dimensional Buddha, then I'll be impressed. (I tried to produce one myself... it involved overeating.)

[jaguar]

Looks like a bigassed bugeyed alient to me.

[SteveDallas]

When I read the title I though it was some kind of German Buddhist bread. (das Brot = bread auf deutsch)

[Beestie]

Reminds me of a remarkable book I read eons ago called The Tao of Physics which quite successfully united findings in subatomic physics and quantum theory to tenets of Eastern thought and mysticism.

The fractal Buddha is a perfect metaphor for a section of the book that discusses the cosmic background radiation leftover from the big bang and the Dance of Shiva.

According to Amazon, the book (published in '75) still has a cult following.

[xoxoxoBruce]

Originally posted by jaguar
Looks like a bigassed bugeyed alient to me.

You seem to have an obsession with big asses today.

[linknoid]

Originally posted by Undertoad

What's the Mandelbrot Set, you ask. Here's a quick introduction to it. It's hard math... basically, it's a mathematical function that is a fractal. What's a fractal, you ask? You ask some damn difficult questions don't you? I didn't major in math!

I majored in math, and to be honest, that's not hard math (and from the viewpoint of a mathematician, it's not even vaguely hard math, you don't even want to know what hard math is). The level of understanding you need to comprehend the mathematics of it is about at Algebra 2 in high school. The mathematics that create these fractals have been around for centuries, but there was never any way to discover fractals until the invention of computers, doing the calculations for one image like that would take the lifetimes of thousands or maybe even millions of people.

If you want a shorter explanation of the Mandelbrot set than that link provided, I'll try to explain it as simply as I can:

First you need to understand imaginary and complex numbers. This is the stuff that was covered in Algebra 2. If you don't know what an imaginary number is, stop reading this explanation because it will just confuse you. Basically all the point of the Mandelbrot set is squaring and adding complex numbers over and over:

A complex number has a real and an imaginary part, so when make a graph, you make the X axis correspond to the real part, and the Y axis to the imaginary part. This gives you a complex number for each pixel you graph. Make sense so far?

Then you go through each pixel you want to graph, and call it's corresponding complex number C. Then you take another variable, usually called z, you run it through this repeatedly:

z = z * z + C

You just keep running it through that equation over and over and over again. For most values of C, z will go to infinity. It will just keep gettting bigger and bigger and bigger, each time you do it. That's what you'd expect from "ordinary" real numbers. If you just keep adding and squaring, of course it's just going to keep getting bigger.

But a strange thing happens when you use complex numbers. Most of them do get bigger and bigger forever. But for a small set of them, instead of getting bigger, z just hovers around zero, and it never goes off to infinity. All the numbers that do this belong to the Mandelbrot set.

Not too hard to understand, I hope. The place it really starts to get interesting is when you get to the edge of the set. It's not like there's a smooth solid border between areas that belong to the set and those that don't, it's an infinitely detailed pattern that could never have been discovered without modern computing power.

I haven't read how the guy who created the "Buddha" image chose to represent the Mandelbrot set graphically, but the "standard" way is simply to use different colors to represent how fast each point goes off to infinity.

I'm sure there are better programs out there to graph fractals, but I wrote (in a couple hours) a Mandelbrot set graph generator for a class project back in college, if anyone wants to play with it:

http://linknoid.net/software/Fractals.exe


Regarding the question of what a fractal is, now that is a harder question, and I'm not going to tackle it in length here. I've read Benoit Mandelbrot's "The Fractal Geometry of Nature" where he explains it in great detail, but to sum it up, it's something that occurs in a fractional dimension. It's way beyond the scope of this explanation to explain what that really means, but by dimension, I'm referring to the same kind of dimensions as when you talk about a drawing or painting being in 2D, or the 3D of real objects, that kind of dimension. And a fractal is something that isn't 2D, or 3D, or even 1D, but something that falls in between, a fractional dimension.

OK, well, I can't help myself, just a short explanation:

If I ask you, how long is the coastline of England, you could get out a map and try to measure it with a ruler. But since you can't measure all the cracks and crevices with any detail, you'll just get a rough estimate. If you go there and drive along the coast, you'll get a larger number because you'll wind around a lot, making it longer. As you get more and more accurate in measuring the length of the coastline, you keep finding it gets longer and longer, until eventually you'll say it's infinitely long. But how can it be? It's not really inifinite in length. And if I try to measure the length in 2 dimensions, you would say it doesn't have any, because a measurement of length doesn't have any "area". So fractal geometry came to the rescue. It has a "length", but not in one dimension. The length exists in a fractional dimension. And fractals were created to try to describe these fractional dimensions.

Many fractals are artificial mathematical constructs, like the Mandelbrot set, but many have been found that describe things in nature, like the structure of a snowflake, or the shape of a cloud, etc. If you try to measure the length of the edge of the Mandelbrot set, as you zoom in, it grows in length just like the example of the coastline. And that's what defines a fractal. Fractals all exist in fractional dimensions. There are all kinds of fractals, and nature is full of them, but the one thing they have in common is that they are described by fractional dimensions.

I hope that answers your questions, Undertoad.

[xoxoxoBruce]

Linknoid, "hard" as in solid, not "hard" as in difficult.

[MAdMoNKEY]

linknoid,

I think I follow your explanation. So what you are saying is that fractals are finite, totally random patterns from a second dimension located somewhere off the coast of England, but to understand them you have to have taken a mathematics course involving integrals?

[linknoid]

Originally posted by xoxoxoBruce
Linknoid, "hard" as in solid, not "hard" as in difficult.

That makes sense. I guess the context seemed to suggest hard as in difficult, but it makes more sense the way you suggest.

[paranoid]

Originally posted by Undertoad
In computer science we call that recursion, although on reading the Wikipedia entry on fractals I learned that they are not all recursive, and so I don't think I understand fractals exactly. (Paranoid, do you want to take a shot here?)

It seems that another *noid already took the chance, but luckily (or unluckily), he made a few minor errors in attempt to simplify complex ideas enough to be fit for consumption.

First, we need to understand that there are different dimensionalities (i.e. ways to determine the dimensionality of an object). The simpliest one is topological. A line is 1-dimensional, a surface is 2-dimensional, a body is 3-dimensional and so on, going further up is easy. Let's also add that a countable set (i.e. a collection of points that can be counted like 1, 2, 3, .... ; it can be infinite) is 0-dimensional and an empty set is -1-dimensional.

There are also other ways to define the dimensionality. The one we need is called Hausdorff-Besikovitch dimensionality (some mathematicians use Minkovsky dimensionality to define fractals instead, which usually, but not always, gives the same result, but we will not go that deep). It's extremely simple. Take a square (or cubic, etc.) grid, draw your object and count how many squares it crosses. Now scale the grid (change the size of the squares, their total number changes accordnigly). Calculate the same. Now look at the dependence between the size of the square edge (d) and the number of squares that contain some points of your object (N). Try it with any curve drawn on a piece of paper. Usually N is (not exactly, of course, but if you decrease the d, the limit will be the x, but let's not go into that) proportional to 1/d^x (d to the power of x). x is the Hausdorff-Besikovitch dimensionality (of course, this is not the strict definition).

If we are talking about simple curves on a plane (or in space, it doesn't matter), then x will be 1, if we are talking about 2-dimensional figures on a place (or in space, etc.), x will be 2. But it turns out, that there are such "curves" that have fractional Hausdorff-Besikovitch dimensionality (how does the number of squares change), even though their topological dimensionality (is it a curve or a surface or a body) is integer. These things are fractals. It's often said that something is a fractal only if its Hausdorff-Besikovitch dimensionality is <B>greater</B> than its topological dimensionality and if it is less, it's called a Cantor set, but it's pretty much the same thing.

The characteristics of a fractal are self-similarity (not necessarily), discontinuity and fractional dimensionality.

Discontinuity is basically what separates them from normal figures, that they have no "defined" shape. Self-simularity is a common thing (Mandelbrot defined fractals in some of his papers based on self-simularity and the name fractal may actually come from latin "fractio" ("fractus"?), which hints at self-simularity. But there are things which are not self-similar, but are very much like fractals, so we usually call them fractals too. You can easily imagine them if you first imagine a simple regular (algebraic) fractal, but then remove the requirement of regularity and understandability. Basically that would be some crazy set of points confined in a square. If you add enough points and if they are crazy enough, they will form a non-regular fractal. This isn't hard math, though. As for fractional dimensionality, this is usually true (and some people [wrongly] define fractals as objects with fractional dimensionality), but not always. For example, Peano curve has a H-B dimensionality of exactly 2 and topological dimensionality of 1. It is a fractal.

These are some funny things to ponder, but the real question that only masters could answer is whether Budda has a fractal nature?

[Troubleshooter]

*shivering*

I see math people...

[wolf]

I'm trying to ignore that to try to retain my "oooh pretty!" wonderment at the results of all that nasty math.

I have a high degree of math anxiety.

It's why I am not a computer programmer. Yes, it was THAT bad that it changed my career path. Which has probably turned out to be not totally a bad thing, since it's hard to outsource crazy people. The other countries tend to have enough of their own that they are trying to outsource them to us. (I am not kidding. We have had a sharp increase in crazy Asians over the last couple of years. Mainly females, shipped to the US by families as part of arranged marriage deals. The girls were already crazy, the arranged husbands or their parents didn't get that particular tidbit during the negotiations.)

[linknoid]

Originally posted by paranoid

It seems that another *noid already took the chance, but luckily (or unluckily), he made a few minor errors in attempt to simplify complex ideas enough to be fit for consumption.

Part of it was trying to simplify it enough for public consumption (like the ideas of what in the world "fractional dimension" means), part of it was stuff I forgot (like the difference between a Cantor set and a fractal) because it was so long ago, and a larger part of it is stuff I never learned or never understood thoroughly. :-)

I think my explanation was probably confusing enough, I doubt anyone who hasn't studied "higher" math will understand what you're talking about. ;-)

[xoxoxoBruce]

It's as clear as mud, but it covers the ground.