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aforsy
the leaping penguin
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Joined: 13 Jul 2004
Posts: 653
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 Posted: 30 Jan 2005 04:16:36 pm Post subject: |
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we're talking about complex coordinates in trig now, and my teacher mentioned fractals, but she made it pretty clear we wouldn't be making any, or even talking about them much, so could someone please explain to me what the principles of fractal generation are?
please don't tell me what they are, i already know, and i know you use iterations and complex numbers (at least, for the really good ones, that is)
thanks a lot |
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alexrudd
pm me if you read this
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Joined: 06 Oct 2004
Posts: 2335
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| Posted: 30 Jan 2005 09:06:29 pm Post subject: |
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The only thing I ever learned formula wise is that the simplest fractal would be x=x+c, except the equals goes both ways. This fractal looks stupid, as it is only a curve. But I think if you made the equation more complicated, you might get a decent fractal.
Ugh, my explanation is pretty crappy. Try Wikipedia |
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DarkerLine
ceci n'est pas une |
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Joined: 04 Nov 2003
Posts: 8328
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| Posted: 31 Jan 2005 06:31:26 pm Post subject: |
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| You have a formula z=f(z). All complex numbers z for which the formula converges are in the fractal; the ones for which it diverges aren't. |
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aforsy
the leaping penguin
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| Posted: 01 Feb 2005 06:43:30 pm Post subject: |
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i don't understand what that means, though. sorry.
i don't understand how to graph a complex number, i guess is part of the problem. but besides that, i don't get what to do. do i just pick an equation? |
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Ray Kremer
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Joined: 16 Feb 2004
Posts: 237
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| Posted: 02 Feb 2005 11:35:36 am Post subject: |
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The only way I've ever seen to graph a complex number is to use the x-axis as the real portion and the y-axis as the imaginary portion. In other words, plot points using the ordered pair (a,b) for a+bi.
It's more like a 2D number line since plotting points is all you can do, I don't think you could do any kind of function graphing using this scheme.
Last edited by Guest on 02 Feb 2005 05:40:18 pm; edited 1 time in total |
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alexrudd
pm me if you read this
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| Posted: 02 Feb 2005 05:58:17 pm Post subject: |
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Sir Robin wrote: You have a formula z=f(z). All complex numbers z for which the formula converges are in the fractal; the ones for which it diverges aren't.
Could you please explain that? I've always wanted to know more about fractals, and most of my teachers don't even know what they are.... I know what complex numbers are, just not anything related to graphing them.
Also, did you edit Ray Kremer's post or is that a bug? |
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sgm
Calc Guru
Joined: 04 Sep 2003
Posts: 1265
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| Posted: 02 Feb 2005 06:30:24 pm Post subject: |
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alexrudd wrote: Sir Robin wrote: You have a formula z=f(z). All complex numbers z for which the formula converges are in the fractal; the ones for which it diverges aren't.
Could you please explain that? I've always wanted to know more about fractals, and most of my teachers don't even know what they are.... I know what complex numbers are, just not anything related to graphing them.
Read this:
http://www.programmersheaven.com/zone10/cat101/807.htm |
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alexrudd
pm me if you read this
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Joined: 06 Oct 2004
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| Posted: 02 Feb 2005 08:02:17 pm Post subject: |
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| Thanks for the link. Unfortunately, I don't know PASCAL and it appears that the file has lost some information (images, etc). But I get the gist of what is being said. Would this be possible to do in BASIC, or would it be too slow? |
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sgm
Calc Guru
Joined: 04 Sep 2003
Posts: 1265
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| Posted: 02 Feb 2005 08:07:19 pm Post subject: |
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| You might want to try deleting the "807.html" and seeing if that gets you anywhere... |
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Weregoose
Authentic INTJ
Super Elite (Last Title)
Joined: 25 Nov 2004
Posts: 3976
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| Posted: 03 Feb 2005 05:05:19 pm Post subject: |
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[font="courier new"]PROGRAM:FRACTAL
FnOff
ClrHome
ClrDraw
GridOff
AxesOff
PlotsOff
‾.2899→Xmin
.2→Xmax
.2182172→Ymin
.7→Ymax
.2+.5[font="times new roman"]i→Z
0→dim(L1
L1→L2
For(T,1,E2
real(Z→L1(T
imag(Z→L2(T
Z²+.2+.5[font="times new roman"]i→Z
End
Plot1(xyLine,L1,L2,·
DispGraph
Zooming into the center of this equation shows that the "tunnel" is theoretically the same throughout. Would this be considered a fractal?
Last edited by Guest on 03 Feb 2005 05:30:34 pm; edited 1 time in total |
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Weregoose
Authentic INTJ
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Joined: 25 Nov 2004
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| Posted: 04 Feb 2005 12:12:36 pm Post subject: |
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Please excuse the double post.
I've been working on a way of finding the focal point of a recursive equation...
0=1-a/b-b
...where "a" is the result of the first iteration (in the post above, when Z=0), and "b" is the result of an infinite number of iterations (i.e., the focal point, never reached; it acts as a sort of positional asymptote).
Example:
(-.02559194298238+.47565417114538i)→Z
1-(.2+.5i)/Z-Z
0
This seems to work just for this particular equation (Z²+.2+.5i), of what I've tested so far. I don't know where else it might apply... I used this to center the graphing window in order to make zooming a breeze, but I feel that it may have some other purpose as well. Well, what that is, I'm not quite sure.  |
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BarrenSoul
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Joined: 22 Dec 2004
Posts: 189
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| Posted: 04 Feb 2005 01:40:32 pm Post subject: |
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I believe this has been done and there is already a developed method (it was used to prove that the lorenz attractor will never stray off to infinity nor hit the focal points) but it's probaby different for every equation :)
Last edited by Guest on 04 Feb 2005 01:40:58 pm; edited 1 time in total |
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aforsy
the leaping penguin
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Joined: 13 Jul 2004
Posts: 653
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| Posted: 05 Feb 2005 04:38:58 pm Post subject: |
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Supergoose wrote: Zooming into the center of this equation shows that the "tunnel" is theoretically the same throughout. Would this be considered a fractal?
actually, if you zoom three times, you're at the end of the tunnel.
i've seen this image as a geometry construction, so i don't think it's a fractal quite yet. if you restructured it so that it re-generates the tunnel each time the user zooms in, then it would be a fractal of sorts, i think, but not exactly, since not every part has the same shape as the structure as a whole.
EDIT: i think you can do this by just changing the For( loop start and end points (T, that is) for whatever section of the tunnel you're generating.
EDIT EDIT: it works okay, but you can't just zoom straight in or it will be off center. you have to adjust the window settings accordingly.
Last edited by Guest on 05 Feb 2005 04:53:55 pm; edited 1 time in total |
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Weregoose
Authentic INTJ
Super Elite (Last Title)
Joined: 25 Nov 2004
Posts: 3976
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| Posted: 05 Feb 2005 05:42:42 pm Post subject: |
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Quote: if you zoom three times, you're at the end of the tunnel.
That's true, since the code doesn't exactly repeat itself an infinite number of times.  I'll take you up on that For( loop idea, however.
Quote: you can't just zoom straight in or it will be off center. you have to adjust the window settings accordingly.
Try these:
[font="courier new"]
-.02559194298238→A
.47565417114538→B
A-.5→Xmin
A+.5→Xmax
B-.5→Ymin
B+.5→Ymax
The above numbers were the coordinates for the focal point I talked about earlier (only accurate to fourteen decimal places, but it works well enough). |
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