2008-09-27, 05:29 PM
[COLOR="Red"]I've seen quite a few of these interesting things...and I wonder if anyone else has seen one.
If you don't know what a fractal is, here:
A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
Such as this one I found...[/COLOR]
[COLOR="Blue"]
When creating a fractal image, we need to generate the numbers that will go into it. This is by far the simplest I've ever seen, but it's still fun if you want something to do.
To produce the fractal, we need to generate numbers by feeding them into a box, taking what comes out, and feeding it back into the same box. This is called an iterative function.
Start Number D ~> | D x (1-D) | ~> New value of D
Choose any decimal number D between 0 and 1, and feed it into the box. For example, the starting point for D might be 0.6. The rule for producing the new value for D is to take the current value of D and multiply it by (1-D). In this case, 0.6 x 0.4 = 0.24.
Now feed this new value of D, 0.24, back into the box, to create the next value of D: 0.24 x (1-0.24) = 0.1824. Repeat this a few times, and D will rapidly close in on 0, and this will happen no matter what number you start with. Nothing too exiting so far.
Things get much more interesting, however, if you add another box, K, into the system.
Start Number D ~> | D x (1-D) | ~> multiply by K ~> New value of D
K is like a control button on this system, so let's see what happens to the final value of D as K is increases.
We already know that if K is 1, D always decays to 0. Suppose instead that K is 2, and you choose a starting value of D as 0.6, like last time.
[COLOR="Black"]First time through the system:
0.6 x 0.4 x 2 = 0.48
...feed this back through to give...
0.48 x 0.52 x 2 = 0.4992
...feed this back to give...[/COLOR]
0.4992 x 0.5008 x 2 = 0.49999...
After just three loops through the system, it's already becoming apparent that D is going to end up at 0.5. What's more remarkable is that, if K is 2, whatever value of D you start with between 0 and 1, it will always end up at 0.5, and it will get there pretty quickly too. Before going any further, you might want to try this for yourself using a different starting number for D. When you're finished with that...
What will happen if K is 2.5? D always ends up at 0.6, whatever value of D you start with.
If K is 3, however, something odd will happen. D eventually ends up oscillating between two numbers, 0.669 and 0.664 Oscillating? How can this be? It's mysterious, and only the start of it.
When K is 3.47, D always ends up cycling around four numbers, 0.835, 0.479, 0.866, and 0.403. From here, as K increases by small increments the quantity of numbers in D's final cycle doubles with rapidly increasing frequency. First D oscillates between eight numbers, then 16, 32, and so on. This process of doubling is called Bifurcation. Finally,when K gets close to 4, there seems to be no pattern at all. D just leaps in a seemingly random fashion from one number to another, never settling anywhere.
Value of K = Final Value of D
1 = 0
2 = 0.5
2.5 = 0.6
3 = Oscillating between 0.669 and 0.664
3.47 = Oscillating between 0.835, 0.479, 0.866, and 0.403
Close to 4 = "We have total mayhem, Captain."
Here at last, are some numbers to plot on a chart, though to produce an accurate diagram you'd need to examine every value of K between 1 and 4. This is because between apparent areas of randomness, there will occasionally be tiny zones where the number of oscillations drops to a small number again - for example, at 3.74, there are five numbers in the oscillation.[/COLOR]
[COLOR="Red"]
=x I don't know how to show a chart on here, so if anyone PM's me instructions, I'll show one of my own. If you try to do it yourself, however, you should make it with D on the y-axis, and K on the x, Since D only goes up to 1, make that 1 big, very big. K only needs to go to 4, so that can be whatever size you want to make it.
What you should end up with, is a line starting at 0 for D and 1 for K, rising to 0.5 for D at 2 for K, 0.6 D and 2.5 K, and almost flat until you get to 3. Once you're there, it'll start splitting, and make an arch, like half a circle, with the midpoint of that half at 3. It continues as one arch until you get to 3.47, where each end of the arch splits in two, making two arches. Every so often after that, the arches will split, again and again, until you get close to 4, at which time total Chaos occurs.
Tell me if you find this interesting. ;o[/COLOR]
If you don't know what a fractal is, here:
A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
Such as this one I found...[/COLOR]
[COLOR="Blue"]
When creating a fractal image, we need to generate the numbers that will go into it. This is by far the simplest I've ever seen, but it's still fun if you want something to do.
To produce the fractal, we need to generate numbers by feeding them into a box, taking what comes out, and feeding it back into the same box. This is called an iterative function.
Start Number D ~> | D x (1-D) | ~> New value of D
Choose any decimal number D between 0 and 1, and feed it into the box. For example, the starting point for D might be 0.6. The rule for producing the new value for D is to take the current value of D and multiply it by (1-D). In this case, 0.6 x 0.4 = 0.24.
Now feed this new value of D, 0.24, back into the box, to create the next value of D: 0.24 x (1-0.24) = 0.1824. Repeat this a few times, and D will rapidly close in on 0, and this will happen no matter what number you start with. Nothing too exiting so far.
Things get much more interesting, however, if you add another box, K, into the system.
Start Number D ~> | D x (1-D) | ~> multiply by K ~> New value of D
K is like a control button on this system, so let's see what happens to the final value of D as K is increases.
We already know that if K is 1, D always decays to 0. Suppose instead that K is 2, and you choose a starting value of D as 0.6, like last time.
[COLOR="Black"]First time through the system:
0.6 x 0.4 x 2 = 0.48
...feed this back through to give...
0.48 x 0.52 x 2 = 0.4992
...feed this back to give...[/COLOR]
0.4992 x 0.5008 x 2 = 0.49999...
After just three loops through the system, it's already becoming apparent that D is going to end up at 0.5. What's more remarkable is that, if K is 2, whatever value of D you start with between 0 and 1, it will always end up at 0.5, and it will get there pretty quickly too. Before going any further, you might want to try this for yourself using a different starting number for D. When you're finished with that...
What will happen if K is 2.5? D always ends up at 0.6, whatever value of D you start with.
If K is 3, however, something odd will happen. D eventually ends up oscillating between two numbers, 0.669 and 0.664 Oscillating? How can this be? It's mysterious, and only the start of it.
When K is 3.47, D always ends up cycling around four numbers, 0.835, 0.479, 0.866, and 0.403. From here, as K increases by small increments the quantity of numbers in D's final cycle doubles with rapidly increasing frequency. First D oscillates between eight numbers, then 16, 32, and so on. This process of doubling is called Bifurcation. Finally,when K gets close to 4, there seems to be no pattern at all. D just leaps in a seemingly random fashion from one number to another, never settling anywhere.
Value of K = Final Value of D
1 = 0
2 = 0.5
2.5 = 0.6
3 = Oscillating between 0.669 and 0.664
3.47 = Oscillating between 0.835, 0.479, 0.866, and 0.403
Close to 4 = "We have total mayhem, Captain."
Here at last, are some numbers to plot on a chart, though to produce an accurate diagram you'd need to examine every value of K between 1 and 4. This is because between apparent areas of randomness, there will occasionally be tiny zones where the number of oscillations drops to a small number again - for example, at 3.74, there are five numbers in the oscillation.[/COLOR]
[COLOR="Red"]
=x I don't know how to show a chart on here, so if anyone PM's me instructions, I'll show one of my own. If you try to do it yourself, however, you should make it with D on the y-axis, and K on the x, Since D only goes up to 1, make that 1 big, very big. K only needs to go to 4, so that can be whatever size you want to make it.
What you should end up with, is a line starting at 0 for D and 1 for K, rising to 0.5 for D at 2 for K, 0.6 D and 2.5 K, and almost flat until you get to 3. Once you're there, it'll start splitting, and make an arch, like half a circle, with the midpoint of that half at 3. It continues as one arch until you get to 3.47, where each end of the arch splits in two, making two arches. Every so often after that, the arches will split, again and again, until you get close to 4, at which time total Chaos occurs.
Tell me if you find this interesting. ;o[/COLOR]
