M Benesi
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« Reply #75 on: March 04, 2011, 07:35:15 AM » |
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« Last Edit: March 04, 2011, 07:37:16 AM by M Benesi »
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fracmonk
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« Reply #76 on: March 04, 2011, 02:44:10 PM » |
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(pix via bunny express)
MB- I think the lines are purposefully rendered under some option to show orbital relationships or something, but I've never been attracted to that sort of thing, and could be extremely wrong in guessing about it.
In the 1st pic below, an equivalent of the Fiegenbaum point from M is shown from the index set I'd been discussing. It's followed by the whole Julia set, and then a magnification of its center, z=1/c, in both cases.
For MY explanation of why the c=1 Julia set takes the San Marco shape, take the formula and "play computer" with it. It mimicks the behavior in M. Of its surroundings, more controversial, maybe the cardioid should be viewed as a distorted bulb.
Real world events will probably put some uncertainty into the availability of my next post. Please be patient if I can't respond in a timely way for a while. And feel free to keep things interesting yourselves...
Later!
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« Last Edit: March 04, 2011, 04:27:20 PM by fracmonk »
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Kali
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« Reply #77 on: March 04, 2011, 03:51:45 PM » |
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NORTH AMERICA DESTROYED BY METEOR
Charlie Sheen OK, found clinging to large white rock in Bahamas
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M Benesi
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« Reply #78 on: March 05, 2011, 09:55:52 PM » |
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MB- I think the lines are purposefully rendered under some option to show orbital relationships or something, but I've never been attracted to that sort of thing, and could be extremely wrong in guessing about it.
Hrmm. I don't recall seeing that option in Xaos... but will definitely check it out. I tend to think that it isn't something that simple: I only found the lines at extreme zooms in certain areas of specific fractal formulas, usually very close to the x or y axis, and if I recall correctly, on the 1/(mu+.25) plane. I'll have to try it in a non-Xaos thing like fractint...
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fracmonk
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« Reply #79 on: March 07, 2011, 02:33:56 PM » |
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(pix via bunny express)
MB- I still don't know what they represent exactly. A 1992 book, Fractals: The Patterns of Chaos, by John Briggs did a broad overview of everything remotely fractal at the time. A caption on p.74 had C. Pickover calling them "Mandelbrot stalks", without further explanation.
Strange that back then, my very first rendering of M was a decomposition plot, that traced field lines. It came from a primitive roll-your-own BASIC program made from a recipe in one of those early books. It seems that in these pix, however, the stalks trace something interior to the prisoner sets.
A zoom series from the object I'd been describing begins in the pix below.
Again, timing of my future posts remains unreliable. All things must pass. Some just can't get behind you fast enough...
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« Last Edit: March 17, 2011, 06:26:46 PM by fracmonk »
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fracmonk
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« Reply #80 on: March 08, 2011, 02:33:03 PM » |
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(pix via bunny express)
Below, the zoom continues into what I like to call a "super-paisley zone", where Julia-like blobs become less "shapeless" than in other regions of the set.
This set is probably the simplest expression (this one contains only M2 minis) of its type (differing from multibrots and the "multiple multibrots" seen first in this thread) as a simply connected set. It has me totally stumped. A working name for them like "Julia-strong index sets" might be appropriate. Your thoughts?
Later!
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« Last Edit: March 08, 2011, 07:17:21 PM by fracmonk »
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fracmonk
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« Reply #81 on: March 09, 2011, 02:24:54 PM » |
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(pix via bunny express)
The zoom continues below, into a spiral that is part tail-of-paisley-shaped blob, and part parent-M2 mini-bristling-with-more-paisley-blobs. Fractals are known to be complicated, and I guess this one does not disappoint...
In the last of these pix, one of the mini's minis begins to become visible.
The journey concludes next time, but when that will be again remains uncertain...
Later!
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« Last Edit: March 09, 2011, 04:17:31 PM by fracmonk »
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makc
Strange Attractor
Posts: 272
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« Reply #82 on: March 13, 2011, 02:00:13 AM » |
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Introducing slit-scan fractalography the ideais to vary some parameter in a way that certain lines will have it constant. here, power p in z = z^p + c is varied according to p = 3 + abs (atan2 (c.y, c.x)) / PI and 4 - abs (atan2 (c.y, c.x)) / PI, i.e we have left-to-right and right-to-left mixes of 4th and 3rd power mandelbrots.
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makc
Strange Attractor
Posts: 272
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« Reply #83 on: March 13, 2011, 02:16:08 AM » |
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same as in prev post, p = 2 + abs (...) i.e. 3rd to 2nd power mix as you go left to right.
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fracmonk
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« Reply #84 on: March 14, 2011, 02:09:33 PM » |
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(pix via bunny express)
makc- pretty neat, but it has discontinuities, as when doing z^.5, etc.
In the 1st pic below, the zoom series begun in previous posts ends. The 2nd shows the whole Julia set for the center coords. of pic 1, itself centered at approximately z=1/c. I guess z=2/c should behave like z=0 does then, and also be critical. The last pic enlarges z=1/c, and the detail corresponds to the location in the index set. Next time, a few more views.
TOTALLY off-topic: On siting nuclear powerplants in earthquake zones ANYWAY in a fear-based economy:
Situation: Seismic warnings indicate a large earthquake is imminent, in about a minute. Should I throw the control rods all the way in without PERMISSION, and get fired for causing a regional blackout after the earthquake doesn't actually happen?
Long-range air quality forecast: airborne radioactive isotope levels higher. To avoid lung cancer, stop breathing.
I have met the enemy, and he is us. -Pogo
In a relatively minor imposition, my jury duty obligation is met, and a CERTIFICATE was issued to that effect, so there should be less restriction on my future posts.
Need to laugh badly? Select ANY newspaper picture, and substitute the caption with this one: "Don't let this happen to you!" It works in almost ALL cases.
Later!
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« Last Edit: March 14, 2011, 06:29:37 PM by fracmonk »
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makc
Strange Attractor
Posts: 272
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« Reply #85 on: March 14, 2011, 02:32:15 PM » |
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using Perlin noise for p in z = z^p + c (edit: now online)
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« Last Edit: March 14, 2011, 03:21:21 PM by makc »
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fracmonk
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« Reply #87 on: March 15, 2011, 01:37:11 PM » |
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(pix via bunny express)
makc- watched your "burning heart" animation. Pretty cool, but I miss the connection with M...
The 1st 2 pix below show subsequent enlargements of z=1/c. The last pic was an almost randomly chosen view, and that finishes what I wanted to show about that object.
I've been working on 2 things lately. One is a coherent collection of the formulae used in this thread in one place, as opposed to "try this, try that, try to remember the file name, etc.", one that would be easier to get a behavioral overview of.
The other is a 4d with 2 real & 2 imag params. I do all my looking in 2d, taking slice views, so I don't have to get involved with 3d programs that probably can't display what I want to show with any fidelity. So I figure on showing some slice views of this very familiar M-like object for you to play with as you will. (Soon as I can)
Later.
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« Last Edit: March 15, 2011, 06:24:47 PM by fracmonk »
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fracmonk
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« Reply #88 on: March 16, 2011, 01:39:18 PM » |
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(pix via bunny express)
However slow it might be, here's another example of the utility of a true, non-trig.-based 3d generator. For all I know, such a thing may exist without my awareness. I can't think of a simpler example:
As an approach to viewing M in 3d (& 4d), we have f(z)->(z+c)(z+d). c and d are equivalently-behaving complex constants, and the critical point is (initial) z=(-c-d)/2. The pix below can be considered as a storyboard plan for an animation. Each has the origin at center, and they show the c parameter plane for fixed d=-2.5, d=-2.25, and d=-2 respectively. You may as well know in advance that for the interval -2.25<d<0, the object is connected in 1 piece, and otherwise split in 2 when d is real, as in the 1st pic. The second shows them beginning to touch. Future posts will continue the series in .25 increments, while a "smooth" animation would require significantly smaller ones, say, .01 per frame. Notice the relationships between values in the interval -2<c<.25 in standard M and values d+.25 here. These are sketchy, hurried renderings, and the last of them is of a pinch point.
More next time.
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« Last Edit: March 16, 2011, 03:25:19 PM by fracmonk »
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makc
Strange Attractor
Posts: 272
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« Reply #89 on: March 16, 2011, 04:26:26 PM » |
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makc- watched your "burning heart" animation. Pretty cool, but I miss the connection with M... the connection is rather straightforward, each frame is actually M-set, but its power varies from pixel to pixel by Perlin noise. if you zoomed into some frame you would see distortions "smoothed out" more as you go deeper, as eventually all pixels on the screen would be calculated with almost same power.
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