 

Undertoad Monday Apr 26 10:54 AM 4/26/2004: Buddhabrot
Quote:
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 twodimensional graph anyway. When they find a function to produce a threedimensional Buddha, then I'll be impressed. (I tried to produce one myself... it involved overeating.) jaguar Monday Apr 26 11:00 AM Looks like a bigassed bugeyed alient to me. SteveDallas Monday Apr 26 11:19 AM When I read the title I though it was some kind of German Buddhist bread. (das Brot = bread auf deutsch) Beestie Monday Apr 26 12:12 PM 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. xoxoxoBruce Monday Apr 26 01:27 PM Quote:
linknoid Monday Apr 26 05:16 PM Re: 4/26/2004: Buddhabrot Quote:
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 Monday Apr 26 06:36 PM Linknoid, "hard" as in solid, not "hard" as in difficult. MAdMoNKEY Monday Apr 26 06:43 PM linknoid, linknoid Monday Apr 26 09:04 PM Quote:
paranoid Tuesday Apr 27 12:14 PM Re: 4/26/2004: Buddhabrot Quote:
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 1dimensional, a surface is 2dimensional, a body is 3dimensional 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 0dimensional and an empty set is 1dimensional. There are also other ways to define the dimensionality. The one we need is called HausdorffBesikovitch 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 HausdorffBesikovitch 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 2dimensional figures on a place (or in space, etc.), x will be 2. But it turns out, that there are such "curves" that have fractional HausdorffBesikovitch 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 HausdorffBesikovitch 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 selfsimilarity (not necessarily), discontinuity and fractional dimensionality. Discontinuity is basically what separates them from normal figures, that they have no "defined" shape. Selfsimularity is a common thing (Mandelbrot defined fractals in some of his papers based on selfsimularity and the name fractal may actually come from latin "fractio" ("fractus"?), which hints at selfsimularity. But there are things which are not selfsimilar, 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 nonregular 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 HB 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 Tuesday Apr 27 02:18 PM *shivering* wolf Tuesday Apr 27 02:23 PM I'm trying to ignore that to try to retain my "oooh pretty!" wonderment at the results of all that nasty math. linknoid Tuesday Apr 27 06:41 PM Re: Re: 4/26/2004: Buddhabrot Quote:
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 Tuesday Apr 27 08:41 PM It's as clear as mud, but it covers the ground.


