The mandelbrot isn't strictly self-similar, which was a bit of a problem when it was devised - it didn't fit the current definition of a fractal. I think I can agree with you that it isn't predictable. However, it is self-similar, not completely, but still. The similarity can also be found in the degree og roughness of the set, which is the same on all scales.
Snowflakes are considered fractals because they do have the same degree of roughness over quite a few levels of magnitude. And they certainly aren't simple euclidian forms.
And of course the question is still being asked weather this really works. I mean, if we can use these simple rules to generate images that look like the original, could it be that the original came to be as it is with similar rules?
Fractals aren't measured just like that, and they are still on a very theoretical level. But they have already helped understanding a lot of stuff.
"we can never create fractals with inifinite dimension"
Really!? Watch me!
First we define a piece of phase space which has infinite dimensions. Then we create an object with one dimension to live in this phase space. So we get coordinates for a point [x, 0, 0, 0, 0...]. Soon enough, we create another one-dimensional object and let it interact with the first one and represent the system with a point in phase space [x, x1, 0, 0, 0...]. Now, with each iteration we let the objects x and x1 move with some rule which makes them depend on each other AND we add another object which depends on all of those who are already there. Now we represent the system [x, x1, x2, x3... xn, 0, 0, 0...] and constantly add an xn and increase n by 1.
And the whole lot creates an attractor with infinite dimensions in infinite phase space!
Fractals can be said to be infinite just the way as pi is infinite. While pi has infinite digits, it never goes beyond - say - 3.15. But all of pi isn't just a finite number.
A true fractal is just a mathematical definition, like pi. You add a little to it, less and less every time, you don't reduce the time as you do it. So a true fractal CAN be said to be infinite, but never exceeds a certain boundary.
But, anyway, infinity is not the interesting point about fractals, but the scale invariance. That is, it remains the same no matter how close you are. Or, in the case of non-strict fractals, has the same degree of roughness, no matter which scale you use. Of course, no natural ocject is a true fractal, instead they can be said to hold for a certain number of magnitudes. If an object is a fractal over just two magnitudes, defining it as fractal really doesn't help much. But objects that remain self-similar over, say, eight magnitudes clearly show a fractal nature, and fractal geometry can help us understand them.
and, of course, no natural object is really infinite.
You find out how the thing works, or evolves. Take a snowflake. You see how it comes to be, you begin with an ice particle, others start hitting it, sticking with it and form a crystal. This is obviously a process which we can set up mathematically. Like, for each iteration, this happens and that adds to the whole snowflake. By using mathematical rules we now use our knowledge of how a snowflake evolves to find its fractal dimension.
Now, fractals are said to be infinite, that is, they have infinite volume, and a self-similarity on all scales. Natural phenomena does, however, not. So no natural object is a TRUE fractal. But obviously, a snowflake IS self-similar, and it remains self-similar over a number of scales. To be a TRUE fractal it would have to be self-similar infinitely.
But anyhow, if a object is irregular, and behaves like a fractal, finding it's fractal dimension (or finding the dimension of the mathematical object representing it) is actually quite useful.
Just think of a fractal as a result of an iterated process. Trees grow leaves. Snowflakes grow, clouds grow, lightnings twist and turn, coastlines get beaten by oceans, etc. The idea of a fractal gives an insight into how such objects come to look like they do.
Actually, it should get a bit easier. Instead of doing a whole lot of complicated euclidian mathematics, which, by the way, never gives more than an estimate, and maybe iterating a VERY complex algorithm to get a closer approximation, you just get the fractal dimension, and the rest comes for free!
Just imagine calculating the space of a snowflake. You begin with dividing the while lot into squares, triangles, circles, etc, calculate each one, and add together, or MELT IT and thereby destroying it. Or you could find the fractal dimension, which, by the way, tels you a WHOLE LOT more than it's mere volume and accept that its volume is infinite (it really isn't, but who's bothering to find THAT out, anyway?).
Ever heard of chaos theory?
The weather system has shown to be chaotic in it's worst sense. That, for example means that small effects can have HUGE consequences, and not only huge, but highly unpredictable.
Now it's true that literally EVERYTHING on our little planet affects the weather in some way or another producing the apparent randomness of the whole lot. So if we invent technology tochange it, it wouldn't be of any consequence. I mean, some weather machine is just as likely to produce a catastrophe as a buttefly in Mexico. So why not, we could maybe control immediate catastrophes if we spot them in time (of course it's impossible to predict such things as tornados).
But then, how can we be sure that any modifications of the weather will turn out as intended? It's not "well, now we press this button and this happens", it's rather "now we press this button, and... gee, who would have thought!".
Anyway, it will be very interesting to see how this thing turns out.
"Intel does not support all gcc language extensions; while it has been used to compile the Linux kernel and other free software projects, it is not a drop-in replacement for gcc."
found at the bottom of http://www.coyotegulch.com/reviews/intel_comp/inte l_gcc_bench2.html
Re:The PROOF that google is evil
on
Google vs. Evil
·
· Score: 1
Since 86597 has been turned backwards twice the number of the great sin 12 should be 21.
Thus 3541 becomes 3541 + 9 = 3550, which is ABSOLUTLEY NOT dividible by the beast number 666.
Therefore google is not necessarily evil.
... I know this woman who was getting pretty pissed becasue of all the spam she got.
So she started to reply, threatening to sue the companys in question. And it worked! Just make it soundlike you're a lawyer and know what you're talking about.
Yeah, but years have an annoying tendency to slip by and disappear. We'll just have to stick with an approximation worth 2 significant digits, so it so until the year 2005, it will be quantum physics 100th birthday.
Thank you! It's pretty simple really. I got obsessed with circles and spirals for a while and got the simple circle algorithm. Using an two-dimensional array it's quite easy to emulate a drawing program, where you just assign '*' or something to array[x][y] and pick a location for x and y.
To draw a circle you use sin() and cos(), to get the coordinates og a point in the ring. the functions take radians as arguments, opposed to degrees. 2pi == 360
Then you multiply the result with a number, that number is the radius of the ring. Keep it constant to draw a circle, add a small value to it to get a spiral. So here's a code to explain this:
double i; int radius = 50;//pixels for(i = 0; i < 2*pi; i+=0.01) drawPoint(sin(i)*radius, cos(i)*radius);
I've also got the algorithms for sin() and cos() if you want:), though they come with math.h
Slashdot Mirror
The mandelbrot isn't strictly self-similar, which was a bit of a problem when it was devised - it didn't fit the current definition of a fractal. I think I can agree with you that it isn't predictable. However, it is self-similar, not completely, but still. The similarity can also be found in the degree og roughness of the set, which is the same on all scales.
Snowflakes are considered fractals because they do have the same degree of roughness over quite a few levels of magnitude. And they certainly aren't simple euclidian forms.
And of course the question is still being asked weather this really works. I mean, if we can use these simple rules to generate images that look like the original, could it be that the original came to be as it is with similar rules?
Fractals aren't measured just like that, and they are still on a very theoretical level. But they have already helped understanding a lot of stuff.
"we can never create fractals with inifinite dimension"
... xn, 0, 0, 0...] and constantly add an xn and increase n by 1.
Really!? Watch me!
First we define a piece of phase space which has infinite dimensions. Then we create an object with one dimension to live in this phase space. So we get coordinates for a point [x, 0, 0, 0, 0...]. Soon enough, we create another one-dimensional object and let it interact with the first one and represent the system with a point in phase space [x, x1, 0, 0, 0...]. Now, with each iteration we let the objects x and x1 move with some rule which makes them depend on each other AND we add another object which depends on all of those who are already there. Now we represent the system [x, x1, x2, x3
And the whole lot creates an attractor with infinite dimensions in infinite phase space!
A fractal refrigerator, cool! So, when you put something into it, does it grow spirals and stuff?
Fractals can be said to be infinite just the way as pi is infinite. While pi has infinite digits, it never goes beyond - say - 3.15. But all of pi isn't just a finite number.
A true fractal is just a mathematical definition, like pi. You add a little to it, less and less every time, you don't reduce the time as you do it. So a true fractal CAN be said to be infinite, but never exceeds a certain boundary.
But, anyway, infinity is not the interesting point about fractals, but the scale invariance. That is, it remains the same no matter how close you are. Or, in the case of non-strict fractals, has the same degree of roughness, no matter which scale you use. Of course, no natural ocject is a true fractal, instead they can be said to hold for a certain number of magnitudes. If an object is a fractal over just two magnitudes, defining it as fractal really doesn't help much. But objects that remain self-similar over, say, eight magnitudes clearly show a fractal nature, and fractal geometry can help us understand them.
and, of course, no natural object is really infinite.
You find out how the thing works, or evolves. Take a snowflake. You see how it comes to be, you begin with an ice particle, others start hitting it, sticking with it and form a crystal. This is obviously a process which we can set up mathematically. Like, for each iteration, this happens and that adds to the whole snowflake. By using mathematical rules we now use our knowledge of how a snowflake evolves to find its fractal dimension.
:-)
fractical
Now, fractals are said to be infinite, that is, they have infinite volume, and a self-similarity on all scales. Natural phenomena does, however, not. So no natural object is a TRUE fractal. But obviously, a snowflake IS self-similar, and it remains self-similar over a number of scales. To be a TRUE fractal it would have to be self-similar infinitely.
But anyhow, if a object is irregular, and behaves like a fractal, finding it's fractal dimension (or finding the dimension of the mathematical object representing it) is actually quite useful.
Just think of a fractal as a result of an iterated process. Trees grow leaves. Snowflakes grow, clouds grow, lightnings twist and turn, coastlines get beaten by oceans, etc. The idea of a fractal gives an insight into how such objects come to look like they do.
and check out my fractal program!
Actually, it should get a bit easier. Instead of doing a whole lot of complicated euclidian mathematics, which, by the way, never gives more than an estimate, and maybe iterating a VERY complex algorithm to get a closer approximation, you just get the fractal dimension, and the rest comes for free!
Just imagine calculating the space of a snowflake. You begin with dividing the while lot into squares, triangles, circles, etc, calculate each one, and add together, or MELT IT and thereby destroying it. Or you could find the fractal dimension, which, by the way, tels you a WHOLE LOT more than it's mere volume and accept that its volume is infinite (it really isn't, but who's bothering to find THAT out, anyway?).
Ever heard of chaos theory? The weather system has shown to be chaotic in it's worst sense. That, for example means that small effects can have HUGE consequences, and not only huge, but highly unpredictable. Now it's true that literally EVERYTHING on our little planet affects the weather in some way or another producing the apparent randomness of the whole lot. So if we invent technology tochange it, it wouldn't be of any consequence. I mean, some weather machine is just as likely to produce a catastrophe as a buttefly in Mexico. So why not, we could maybe control immediate catastrophes if we spot them in time (of course it's impossible to predict such things as tornados). But then, how can we be sure that any modifications of the weather will turn out as intended? It's not "well, now we press this button and this happens", it's rather "now we press this button, and... gee, who would have thought!". Anyway, it will be very interesting to see how this thing turns out.
"Intel does not support all gcc language extensions; while it has been used to compile the Linux kernel and other free software projects, it is not a drop-in replacement for gcc."
e l_gcc_bench2.html
found at the bottom of http://www.coyotegulch.com/reviews/intel_comp/int
Since 86597 has been turned backwards twice the number of the great sin 12 should be 21. Thus 3541 becomes 3541 + 9 = 3550, which is ABSOLUTLEY NOT dividible by the beast number 666. Therefore google is not necessarily evil.
... I know this woman who was getting pretty pissed becasue of all the spam she got. So she started to reply, threatening to sue the companys in question. And it worked! Just make it soundlike you're a lawyer and know what you're talking about.
Yeah, but years have an annoying tendency to slip by and disappear. We'll just have to stick with an approximation worth 2 significant digits, so it so until the year 2005, it will be quantum physics 100th birthday.
Of course we could just get a butterfly to flap it's wings a couple of times inside the computer. That oughta do it. :)
Thank you! It's pretty simple really. I got obsessed with circles and spirals for a while and got the simple circle algorithm. Using an two-dimensional array it's quite easy to emulate a drawing program, where you just assign '*' or something to array[x][y] and pick a location for x and y.
//pixels
:), though they come with math.h
To draw a circle you use sin() and cos(), to get the coordinates og a point in the ring. the functions take radians as arguments, opposed to degrees. 2pi == 360
Then you multiply the result with a number, that number is the radius of the ring. Keep it constant to draw a circle, add a small value to it to get a spiral. So here's a code to explain this:
double i;
int radius = 50;
for(i = 0; i < 2*pi; i+=0.01)
drawPoint(sin(i)*radius, cos(i)*radius);
I've also got the algorithms for sin() and cos() if you want
Try Makefiles! :)
And, just because I'm a sixteen year old f00l,
I'd like to paste this annoying little program:
double pi = 3.14159265358979323846;
int matrix[25][30];
double radius = 2;
for(double d = 0; d < 6*pi; d+=0.05) {
matrix[12 + (int)(cos(d)*radius)]
[15 + (int)(sin(d)*radius)] = '.';
radius += 0.03;
}
for(int i = 0; i < 25; i++) {
for(int j = 0; j < 30; j++)
printf((matrix[i][j] == '.') ? ". " : " ");
printf("\n");
}
Don't be hard on yourself, at least you can hack :)
Isn't there a library I code in C, is there hope for me yet? Can't I just include sex.h or something?
nothing here
When this thing is ready, can I get a copy of you guys? I'd like it bitrate 192, I don't like quality loss :)