New Mandelbrot Variation (Beetle)

[kaludix]

I was experimenting with using the iteration number in the fractal equation and discovered the following interesting Mandelbrot variation:

EDIT 8/10/2016: Equation should be

The overall structure is as follows:


The equation is straightforward to calculate using the following innermost loop:

Code:

iteration = iteration + 1
If iteration Mod 2 = 1 Then
     xtemp = -x * x + y * y + x0
     y = -2 * x * y + y0
Else
     xtemp = x * x - y * y + x0
     y = 2 * x * y + y0
End If
x = xtemp

I’ve been calling this fractal the Beetle fractal and incorporated it into my code Fraqcel along with perturbation and series approximation routines.  The resulting fractal has some interesting characteristics.  A couple zoomed views are as follows:




Both the Beetle fractal above and the Mandelbrot set are members of the following more general parameterized fractal family which is also coded into Fraqcel .

EDIT 8/10/2016: Equation should be

The following video shows how the two fractals are connected:
EDIT 8/10/2016: Video updated to display corrected equation
<a href="http://www.youtube.com/v/Elo3wtuQokw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Elo3wtuQokw&rel=1&fs=1&hd=1</a>

I haven’t run across any fractal equations that exploit the iteration number in this way.  If this has been explored elsewhere please let me know.

[quaz0r]

i did a few renders of this one during some of my own tinkering sessions with the mandelbrot equation.  i referred to it as the scarabrot    

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=18367

http://www.fractalforums.com/index.php?action=gallery;sa=view;id=18374

noting also that it looks like neither of us bothered to use a correct critical point for this, instead defaulting to 0, which i think is what causes ones like this to be disconnected.

[kaludix]

I could be wrong but I think the critical point is z=0.  Isn't this were the derivative of the equation with respect to z is zero?

Locations with positive real part appear connected with large chunks that do not escape while areas with negative real part are clearly disconnected.  Your "mandelbrot asteroids" image looks like its from the positive real part area.  I like your coloring on that image.  It highlights the border areas nicely preventing the blobs from standing out too much.

[quaz0r]

thanks   

at the time when i was tinkering i thought, what if i alternate adding and subtracting C.  i guess maybe this is why my beetle faced the opposite direction as yours, where you are alternating the sign the other way.  in any case i thought, this actually just means the equation is (z²+c)²-c which means the actual expanded equation is z⁴+2cz²+c²-c, which i guess would have a non-zero critical point.  im not a mathematician, but this was my thought.

[TheRedshiftRider]

I've seen this set before while experimenting. I think I have a shorter and simpler function for that. I will have a look at what I can find.

Nice find.

[kaludix]

The feedback from quaz0r got me thinking about modifying just the c term.  It didn't occur to me that the result would be the same.  The Beetle fractal remains unchanged but the more general formula rotates in the opposite direction.  Rotating only the c term is more elegant and I believe makes the more general equation easier to calculate.

I'll update my work and report back.

[kaludix]

What I’m calling the Beetle fractal is indeed the same as alternating the sign of the c term each iteration.  The proof is only a few steps:
-- Multiply the equation by (-1)^(-n)
-- Use (-1)^(-n) = (-1)^n
-- The resulting (-1)^n on the left hand side is inconsequential due to subsequent squaring

What is different is the more general parameterization.  Applying the rotation to the c term results in


The following video shows an analogous animation to the initial post.
EDIT 8/10/2016: Video removed.  Calculation performed incorrectly.  See 8/7/2016 post.

Initially I thought the details of this parameterization (c rotation) would be only trivially different to the initial parameterization (z^2 rotation).  Upon zooming in, however, I discovered that applying the rotation to the c term results in less interesting details: no minibrots I could find for example.  I was hoping for something equivalent in beauty because rotating the c term is less computationally expensive.

For the reason discussed above, I left the equations unchanged in my program.  Below is an example of the rich detail found in the z^2 rotation version for k = 2.05.

[quaz0r]

Upon zooming in, however, I discovered that applying the rotation to the c term results in less interesting details: no minibrots I could find

you get correct results and hence minibrots if you use a correct critical point (not using a correct critical point results in incorrect results).  i havent bothered trying it yet with this formula, maybe you can and show us the results 

[kaludix]

The critical point is defined as the value of z that satisfies the following equation:

In all the equations I studied in this thread, the equation above implies the critical point is z = 0.

[quaz0r]

if theres no minibrots, you didnt use the right critical point, is my understanding anyway.

http://www.fractalforums.com/index.php?topic=20898.msg81208#msg81208

When the solutions of f'(z) = 0 include c (the location in the complex plane) you get location dependent critical points which have to be determined for each location. e.g.
z = z² + cz + c
f'(z) = 2z + c = 0
so the critical point is -c/2

so by that logic, i guess for z⁴+2cz²+c²-c you would end up with 4z³+4cz?  which gives you...i dont know, im not a mathematician    wolframalpha says the root of that is  , is that right?

[jdebord]

. There are three roots : . So there are three critical points, two of them depend on c.

[kaludix]

Previous post is correct except the got omitted.  Roots are

this actually just means the equation is (z²+c)²-c

I'm not sure that's true.  Plotting that equation with z₀=0 results in what appears to be the same overall structure except the number of iterations to exceed bailout is different.

[quaz0r]

except the number of iterations to exceed bailout is different

is it that you are counting z²+c as 1 iteration, and z²-c as 1 more iteration?

[kaludix]

I discovered a few issues.
1. In all the equations where n is in the superscript, it should actually be (n+1) to match what is plotted.  Difference is a change in orientation.
2. The general parameterization where the rotation is applied to the c term was incorrectly coded.  Although I intended to use z₀ = 0, I incorrectly applied z₁ = c.

I'll spend some time getting this straightened out and report back.  I'm still hoping the rotation can be applied to the c term since that is computationally less expensive.

Also, the difference in iterations between the 4th order equation (z²+c)²-c and z²±c can be observed by manually calculating the orbits for points on the real axis.  This exercise lead me to discover the issues above.

[kaludix]

Got this straightened out.  Version 2.5 of my program (Fraqcel) applies the rotation to the c-term in the correct manner.  Applying the rotation to the c-term gives similar results as applying to the z²-term and contains the same rich detail.  The final equations are as follows:

Beetle Fractal
     ,     

Rotation Family Fractal
     ,     

The Beetle fractal is not new and consists of just alternating the sign of the c-term with each iteration.  The generalization of the Beetle fractal to the Rotation Family fractal appears to be novel.  I coded both of these fractals into Fraqcel with both perturbation and series approximation routines.  This required some care because .  This weakens the similarities to the Mandelbrot set.

The following video illustrates the parameter sweep of the updated equation:
<a href="http://www.youtube.com/v/dEazbr81CTQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/dEazbr81CTQ&rel=1&fs=1&hd=1</a>