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91
Meet & Greet / I'm back!!
« Last post by chronologicaldot on June 13, 2019, 08:11:39 AM »
Hey everybody!
I'm sure most of you have probably forgotten about me, if you knew me at all. Years ago, I was quite active on DeviantArt (Edit: under username "ABlipinTime") and JWildfire forums. It has been a very long time since then, and though I've sort of peeked in every now and then, I can't say I'm up to speed with what's going on these days. Did manage to find out we're using Discord App now though. Farewell, Slack, you crappy app!

Looking back through my old gallery on DA, I must apologize - I really did post some real crap and never got around to posting my best stuff. If anyone's willing to render some params for me, I'd be happy to share the flame files. :D I use JWildfire, although my version is 2.5 something - haven't downloaded 4 yet. Been busy.

I've been doing 3D stuff more these days, but I'm looking to get back into 2D with the creation of some new software, soon to be announced.

Fun times ahead, hopefully!
93
Menger Fold can be used anywhere it works as a fold (with or without the scaling and adding of constants). It was initially implemented  so that the menger formula could be split into two parts and used in hybrid mode     i.e  slot#1 T> Menger "fold part only" , slot#2   anything you want to test,  slot#3   T>Menger "scale and add constants parts"   Like in formula Menger Middle Mod. 

But you can use T>Menger and T> Octo anywhere for one or more iterations depending on DE quality e.g   pseudo Kleinan,  BoxBulbs. When you break it all down they are just cool folds.

Benesi T1 and T1 Mod  are good too, introducing the "Magic Angle" fold,  and prism shape  introduces 60 deg folds

I nearly always use T>spherefolds and T>Boxfolds , and then use other folds to tweak some variation .

Yep the assymmerty was a surpise to me too.  It shows up with  certain coloring techniques

94
Fractal Mathematics And New Theories / Re: Systems of two real variables
« Last post by pauldelbrot on June 13, 2019, 02:58:21 AM »
There are several Volterra-Lotka Julia fractals in my gallery:

https://fractalforums.org/index.php?action=gallery;su=user;cat=43;u=97

The six images starting with "Superstring Theory" are Volterra-Lotka. The named one has a strange attractor and "Neon Downtown Alpha Centauri" has an invariant circle, or limit circle; both highlighted in neon blue. The bright spots in "The Pleiades Vortex" are a period 7 discrete cyclic attractor. The paler brown haze in the "Smelter" is a strange attractor. "Organic Twist" has holes in the basin of a fixed point; in this case, points in the holes escape, but there are parameters where they'd go to a secondary cyclic attractor instead.
95
Share a fractal / Re: exp(a/z)*z^2+c
« Last post by gerrit on June 13, 2019, 02:50:21 AM »
a=-0.30952, a/2 orbit. This is inside the main cardioblob.
96
Fractal Mathematics And New Theories / Re: Systems of two real variables
« Last post by gerrit on June 13, 2019, 02:14:16 AM »
Embedded Julia set in
\(
x \leftarrow x^2 - y^2 +c_x\\
y \leftarrow ax^2 +2bxy + c_y
 \)
with a=-0.571431, b = 0.886901. Those black minis are smooth (not fractal).
I think the inherent skew, like in the Henon images, gives it a bit of a 3D look.
97
Fractal Mathematics And New Theories / Re: Systems of two real variables
« Last post by Spyke on June 12, 2019, 11:58:48 PM »
According to Peitgen and Richter, in the area in the upper right, the fixed point (1,1) is attractive. Moving left and down, it bifurcates and becomes an attractive invariant circle. This would the teeth, if you are calling the darker regions the teeth, plus perhaps (continuing the analogy) some "gum" in the transition. P&R call the lighter area between the teeth, tongues. I guess they are putting the head on the other side. Anyway, that area has an attractive, finite, periodic orbit, and it is called resonance. Beyond that you get multiple disjoint attracting orbits or orbits and circles, then it transitions to strange attractors, and then escape to infinity.

I kept confusing myself as I typed the about. I had to stop and think about which plane is which. Paul's picture is of the h/p plane (M-like). The dynamical behavior is the dynamics on the x/y plane (J-like), with h & p parameters.


98
Fractal Mathematics And New Theories / Re: Systems of two real variables
« Last post by Spyke on June 12, 2019, 11:25:51 PM »
Paul, have you generated and x/y images for the Discrete Volterra-Lotka?

Peitgen and Richter devote a short chapter to these equations in The Beauty of Fractals. (A thirty three year old book, but still the most referenced book on my bookshelf.) Is this your source? They have eight images, four with h=0.739, p=0.739. and four with h=0.8, p=0.86. The have one, very poor, probably hand drawing two tone image of the h-p plane. Normal coloring should work for the x/y plane. The dynamics have one or two attraction basins, plus escape-to-infinity.
99
Share a fractal / Re: Nested Mandelbrots
« Last post by LionHeart on June 12, 2019, 11:25:22 PM »
Thanks quadralienne.

Nice. Keep 'em coming.
100
This is really off the cuff, I offer no mathematical rigor, but if I were motivated to serious analysis, this would be a starting point.

It seems by that logic, there is no need to single out trig functions.  Anything with a Taylor series expansion can be approximated by quadratic polynomial plus higher order terms. So every critical point looks like z^2+c, plus higher order terms. In very high magnification, small offset from the center of the Taylor series, you ignore the high order terms and everything looks like a mini.

But that only works when you can "ignore high level terms". In the big picture (screen width = 4.0, or whatever is appropriate for your formula), different formulas have very different appearance. Just don't zoom in. (Is there a forest and trees analogy here?)

I suspect that one could argue that a general 2d real system is equivalent to a complex system z = f(z,\( \bar{z} \)). After all, if I have z and \( \bar{z} \) I can break out the real and imaginary parts with simple complex addition. And in this case, the mandelbar shows up in the conjugate analogue of the Taylor series expansion.





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