New(?) variation on the Mandelbrot

[laser blaster]

I was playing around with some formulas, and I stumbled upon a fractal that looks like a distorted M-set, although it has slightly different characteristics. It's not connected, and it looks oddly irregular in places, notably on the two bulbs above and below the main cardioid.

The formula is: Z_new = Z² + Z*C + C

I'm sure someone has already found this before, but I searched the forums and couldn't find it, so I thought I may as well share it.

This formula produces some interesting patterns in places, but it's not as elegant as the M-set, and, as far as I can tell it seems entirely devoid of mini-sets (which is a bummer, as the most interesting deep zoom patterns in the M-set are found in minibrots).

Here's the whole set:


Here are some zoom pics:

[lkmitch]

I think that the variations you're seeing is because you didn't start iterating at the critical point.  It looks like you began with z = 0, but the critical point is z = -c/2.  If you make that change, you'll see that the new set is very similar to the standard Mandelbrot set.  That's because all quadratic functions in z have the same basic dynamics, when begun with the critical point.  If you use your formula beginning at z = 0, that's conceptually similar to using the standard Mandelbrot formula with a starting point different from z = 0.

[Kabuto]

What lkmitch said. A bit more mathematically:

The formula is: Z_new = Z² + Z*C + C

Replacing Z:

Z := Y-C/2

-> Y_new-C/2 = Y² - Y*C + C²/4 + Y*C - C²/2 + C
-> Y_new = Y² - C²/4 + C*3/2

Thus, what you see is basically the mandelbrot set, except that:
* iteration starting point is not 0 but C/2 (explaining why the features are distorted)
* the whole mandelbrot set is warped a bit (C*3/2 - C²/4 instead of C)

I hope that's halfway comprehensible - I don't know your mathematical background.

[laser blaster]

Thank you, that makes sense. You explained it quite well. The one thing I don't understand is this critical point, and why I would start iterating at it. I read the definition of a critical point on Wikipedia, but I don't get the significance of it.

[elphinstone]

Interesting. I never cared about the starting point of the iteration, I Always thought it was somehow irrelevant... thank you Kabuto for your explanations!

Nice pictures anyway!

[rhutson]

The formula is: Z_new = Z² + Z*C + C

There is no longer any need to guess when iterating quadratics using Mandelbrot's method.  They're going to all look "similar".  (See my post here: http://www.fractalforums.com/new-theories-and-research/a-different-view-of-the-mandelbrot-set/ )

With your iteration, plug

z^2 + z*C + C = z - f(z) / f'(z)

into http://www.wolframalpha.com

You are applying Newton's method to this family of functions:



Note that the divisor of the inverse tangent is sqrt(-C^2 + 6C - 1).  The roots of that expression
are 3 ± 2*sqrt(2).  Zoom in around 0.1715728 and 5.828427, and you will find V-ish focal points
similar to the Seahorse Valley around 0.25 on the classic Mandelbrot fractal.  (I haven't found any
programmable Mac OS X software to even confirm this, but I know that they are there.)

[Roquen]

Assuming you have a reasonable GPU, then you can play around using http://www.shadertoy.com or http://glsl.heroku.com/

[hsmyers]

rutson, have you got a mathematica notebook that allows graphing?

--hsm

[rhutson]

rutson, have you got a mathematica notebook that allows graphing?

--hsm

I'm using http://www.wolframalpha.com right now (the $5?/month Pro version) which includes limited Mathematica.  It includes basic plotting such as:   

plot z/(1-z) from z = -10 to 10

[hsmyers]

Well admittedly Mathematica is probably the most expensive fractal view around so I can't say I surprised that use Alpha/Pro instead. I was just hoping you had already  <sigh/> Perhaps if you continue to post with Ultra Fractal code and Element 90 keeps posting Saturn code, exploration will be easy enough to at least get started!

--hsm

[rhutson]

Well admittedly Mathematica is probably the most expensive fractal view around so I can't say I surprised that use Alpha/Pro instead. I was just hoping you had already

I may end up buying / leasing a copy of Mathematica.  I have been exceeding the extended standard computation time with Wolfram|Alpha regularly followed by a suggestion that I purchase Mathematica.    My concern is the amount of time it will take to learn how to use Mathematica.  But then I haven't been able to find programmable fractal explorer for Mac OS X ...

Back to the quadratic recurrence

z_new = z^2 + C*z + C

Just today I generalized Mandelbrot's recurrence to:



Where:
C is Mandelbrot's "constant" element of the complex domain
a,b,q,r,s are elements of the complex domain

(I nearly ran out of variables to use (e and i would be confusing, [i-n] are integers, f and g are functions, c would be confusing, x & y are usually real, etc.))

In the classic Mandelbrot iteration,
q=0, a=1, r=b=0, s=d=1

In laser blaster's iteration,
q=0, a=1, r=b=1, s=d=1

Now here's something very interesting.  Entering

C^q*a*z^2 + C^r*b*z + C^s*d

into Wolfram|Alpha reveals:



Performing my "inverse Newton" method :

C^q*a*z^2 + C^r*b*z + C^s*d = z - f(z) / f'(z)

yielded



Examining the divisor:

4 a d C^(q+s)-b^2 C^(2 r)+2 b C^r-1

also revealed:



"Oh my goodness, it's full of stars."  I thought.

I have already asserted the conjecture that the roots of the divisor in the "inverse Newton" form correspond to a unique seahorse valley cusp which is part of the classic Mandelbrot fractal.  Now I am beginning to grasp why that geometric pattern is repeated "uncountable" times.

[Roquen]

AFAIK: Mathematica Home addition is full featured.

[DarkBeam]

I have seen this variation somewhere. You simply cannot beat Fractint lists

[Nahee_Enterprises]

   With your iteration, plug
          z^2 + z*C + C = z - f(z) / f'(z)
    into  http://www.wolframalpha.com/
    You are applying Newton's method to this family of functions:
           <Quoted Image Removed>
     (I haven't found any programmable Mac OS X software to even confirm this, but I know that they are there.)

   I may end up buying / leasing a copy of Mathematica.  .....  My concern is the amount of time it will take to learn how to
    use Mathematica.  But then I haven't been able to find programmable fractal explorer for Mac OS X ...

Have you ever tried using any of the MAC versions of Terry W. Gintz's software found at Mystic Fractal??  It has a fairly good Formula Parser, capable of handling several functions and complex expressions.
 

[Nahee_Enterprises]

    I have seen this variation somewhere . You simply cannot beat Fractint lists

There are so many formulas available, which have been used within FractInt, that something just like or very similar could be found within the over 8,000 formula collection of OrgForm Compilation.