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Author Topic: inverse gravity law in edge-free universe for added parameter in MSet?  (Read 1536 times)
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kram1032
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« on: September 06, 2012, 05:25:54 PM »

*** First, I introduce the full idea I came up with. Below are actual formulae to play with, so if you want to skip down to that, scroll down ***

I recently played around with the following thought:
Newton's law of gravity states:
F_m1=-F_m2= m1*m2*G* 1/(r.r) * r/norm(r)

Now, the universe is said to be edge-free and the geometry it's estimated macroscopically, usually is a 3-sphere.
This means, that the universe has an (expanding) Radius R and if you go in a straight line, you'll eventually end up at the position you started (edge-freeness)
Since in Newtonian Physics, Gravitational pull is both infinite and unconstrained by speed of light (in relativistic physics it would be limited by speed of light but considering String theory, Gravitons could take shortcuts that might mean, it actually can appear to go faster.), you gotta sum up for infinite distances in both directions.
If you now do the math on one of the grand-circles (straight paths) in that universe, you end up with the following as a sum of the net-force across the entire universe:

|F|=m1*m2*G/(2 R Sin(r/(2 R))²

Now simplifying a bit, the masses and the gravitational constant can be normalized to 1. Furthermore, you can change the Universe's Radius to its Diameter D=2R instead:

1/(D Sin(r/D)²

Inverting gives
(D Sin(r/D)²

And from there you simply introduce the typical constant c.

r = (D Sin(r/D)²+c

This is roughly like an MSet with constrained "universe-Radius" R as additional parameter.

In case you consider all three parameters complex-valued, you end up with the following formulae for the real and imaginary parts of r:
Code:
r=x+yi // standard MSet variable, representing distance between mass-points
D=c+di // Diameter of the "Universe"
C=a+bi // standard MSet constant

//variables to simplify the actual formulae:
sqD = c²+d²
Dpos = 2*(c*x+d*y)/sqD
Dneg = 2*(d*y-c*x)/sqD

reD = c²-d²
imD = 2*c*d

rer = 1/2 * (1- Cos(Dpos)*Cosh(Dneg))
imr = -1/2 * Sin(Dpos)*Sinh(Dneg)

// iterated Functions:

x = reD*rer-imD*imr+a
y = reD*imr+imD*rer+b


(The actual Newtonian law of gravity might end up being more complicated, since you have to integrate over all directions. This is essentially the law applied to a single direction.
Though it behaves correctly in at least the following ways:
- It repeats every 2 pi R, as expected
- In the limit where R goes to infinity, the usual Force is recovered
)
« Last Edit: September 06, 2012, 06:02:43 PM by kram1032 » Logged
Syntopia
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« Reply #1 on: September 07, 2012, 05:52:08 PM »

So, how does it look? Can't you post some images?
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kram1032
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Posts: 1863


« Reply #2 on: September 07, 2012, 06:54:49 PM »

EDIT: before accidentally closing the window, I *did* make screenshots though:
Buddhabrot render: (right click and view image for higher res)

Antibrot render:


Working on it.
Having a real universe-Radius results in really boring images that don't look fractal at all (or maybe if I zoom out a lot or fiddle with the cutoff condition...)
But then I set the Radius to Sqrt(2)-Sqrt(2)i. The Buddhabrot set of that looks almost like if the default anti-Buddhabrot is the full moon and this one is a quater-moon or something.
It's also slightly rotated to the default shape.
The anti-buddhabrot looks almost exactly like the default anti-buddhabrot but with slightly rotated features.

My only problem is that my own program for rendering a Buddhabrot is fairly slow - especially, once the calculation includes any kind of trig-function. I had the Buddhabrot version ready, wanted to render it over-night, but then, in a moment of brain-fart, I shut down the computer >.<

If you want a Mandelbrot or Julia render of it, I don't have a renderer for that readily available. You might want to try that on your own for now. Maybe the escape-time view of real-valued Radii isn't as boring as its orbital-view...
« Last Edit: September 07, 2012, 07:06:38 PM by kram1032 » Logged
kram1032
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Posts: 1863


« Reply #3 on: September 08, 2012, 04:46:23 AM »

Here's an update on the Antibrot. I wish I had a more efficient program...
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kram1032
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Posts: 1863


« Reply #4 on: September 08, 2012, 03:35:14 PM »

Apparently, I had a bug in the appearance of the object, caused by code that's used to help make bilateral sets clear up quicker. I didn't entirely confirm this yet but I based it on my experience regarding that section of the code.
Because a different radius (1+Sqrt(3) i or 2 , 60°) doesn't give anti-buddhabrot-like appearance despite being a buddhabrot render.
Also, it renders much quicker now. (right click and view image for higher resolution)


Note: It almost looks like some sort of non-uniformly stretched and rotated Mset. I suspect the bigger, blurrier blob on the top right to be what in the normal structure would correspond to the "head" and the more detailed but smaller thing on the top left might be what would typically happen in the end.
Though I can't be sure. To find out about that, I'd need to somehow continuously render an animation from R=infinite to R=1+Sqrt(3) i and see what happens.
« Last Edit: September 08, 2012, 03:42:32 PM by kram1032 » Logged
Chillheimer
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« Reply #5 on: October 08, 2017, 01:35:38 PM »

I somehow missed out on this thread. Now those images are down - Kram, do you still have them? mind to repost this in the new forum?
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kram1032
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Posts: 1863


« Reply #6 on: October 08, 2017, 01:54:42 PM »

I fear these are technically lost. (I really should stop using puush for permanent images. I think this was before they announced they'd stop storing images forever though.)

But anyway, I can try rerendering them. The forumla really isn't that difficult though. If you happen to have your own Buddhabrot renderer you should be able to reproduce them.
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Sharkigator
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« Reply #7 on: October 09, 2017, 01:40:20 PM »

I've got a Buddhabrot renderer... (see https://fractalforums.org/other/55/buddhabrot-magnifier-a-realtime-buddhabrot-zoomer/384/)

I adjusted the formula to work with Buddhabrot Mag.
(I hope I did it correctly.)
Code:
_INIT_
complex D;
real dPos;
real dNeg;

real reD;
real imD;

real rer;
real imr;

//adjust these two parameters
D = { sqrt(2), -sqrt(2) };


_LOOP_

dPos = 2*(D.R*Z[n].R + D.I*Z[n].I) / absSq(D);
dNeg = 2*(D.I*Z[n].I - D.R*Z[n].R) / absSq(D);

reD = D.R^2 - D.I^2;
imD = 2 * D.R * D.I;

rer = 1/2 * (1 - cos(dPos)*cosh(dNeg));
imr = -1/2 * sin(dPos) * sinh(dNeg);

Z[n+1].R = reD * rer - imD * imr + C.R;
Z[n+1].I = reD * imr + imD * rer + C.I;


_BAIL_

absSq(Z[n+1]) > 4;

Attachment A is render with D = (sqrt(2), -sqrt(2)};
It looks like the Buddhabrot, but slightly rotated / curved.

Attachment B is render with D = { 2, 2 };
It looks like the Mandelbar, but also slightly rotated.


* renderA.jpg (114.45 KB, 1000x1000 - viewed 131 times.)

* renderB.jpg (90.81 KB, 1000x1000 - viewed 161 times.)
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kram1032
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Posts: 1863


« Reply #8 on: October 09, 2017, 02:04:49 PM »

Looks right to me. I honestly can't remember how they looked like, it's been a while. Try fiddling with the D parameter to see if anything more interesting happens. Note, the larger D is (slightly busy but I'd have to think about whether that's specifically the real part of D or rather the absolute value but I think it's the latter), the more it should approach the bog standard Buddhabrot set. So a smaller D might reveal something more interesting. I think. As said, gotta re-think through this. It's been forever.
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