How Mandelbrot deforms a grid

[Kali]

I wasn't doing too much research on anything lately, neither further improvementes of my previous ideas, but today I tried something and liked the results.
Don't know if this was made before, but I find it quite interesting.

The idea is to plot a grid beign deformed by the Mandelbrot set formula.
I represented the complex plane as a chessboard-like pattern, and this are the results of this pattern altered by the formula after some iterations:














To achieve the chessboard design, I used UF5 with a method of fractional digit extraction as described in another of my posts,

As for now, it's only a simple coloring method I made just for fun. But perhaps I can use this to find properties and relationships in the set that could be useful somehow.

I will try some other things later, maybe different formulas & methods, to see if there are more interesting results.

[Fractal Ken]

Neat idea, Kali! You seem to have done the sort of image manipulation I like, where your input is a picture of a checkerboard.

Here's one with a 1024 x 1024 pixel checkerboard having 64 x 64 pixel squares. I colored the Mandelbrot set's interior using two "orbit traps" (in particular, minimum distance from the x and y axes) to select rows and columns from the input image.

[Kali]

Thanks Ken... you also did a good work with that image!
And I'm surprised you almost always have a similar selfmade fractal in your archives!  

As I told you before, I really like your technique and I will be trying soon to do something like that.
But this is different, because I didn't use an existing input image, and no orbit traps were used neither.

This is what I've done:

1. Iterate Z=Z*Z+C the desired number of times (with no bailout condition).

Then, being X the real part of Z, Y the imaginary part:

2. multiply X,Y with a scale factor that will set the size of the pattern:

X=X*scale
Y=Y*scale

3. extract the fractional part of X,Y into DX,DY:

DX=abs(X-floor(X))
DY=abs(Y-floor(Y))

4. evaluate DX,DY to determine if painting white or black (1 or 0):

if (dx < .5 and dy < .5) or (dx > .5 and dy > .5)
      color=1
else
      color=0
endif

Off course this doesn't include any antialias filter, and a lot of aliasing occur in the areas that the patterns become very small. The images I posted were generated with UltraFractal using the built-in antialias function. There's still some visible aliasing, tough.

I should used a bailout condition because after some iterations, the background becomes solid, as the values gets very large and no longer being handled by Ultrafractal. But I wanted to see the grid deformation of the "outside" when doing few iterations.

This is 5 iterations, with a small scale value for the pattern (1e-10)... take a look:

[Fractal Ken]

And I'm surprised you almost always have a similar selfmade fractal in your archives!  

Actually, I just rendered it to post in this discussion. It was really easy.

As I told you before, I really like your technique and I will be trying soon to do something like that.
But this is different, because I didn't use an existing input image, and no orbit traps were used neither.

This is what I've done:

1. Iterate Z=Z*Z+C the desired number of times (with no bailout condition).

Then, being X the real part of Z, Y the imaginary part:

2. multiply X,Y with a scale factor that will set the size of the pattern:

X=X*scale
Y=Y*scale

3. extract the fractional part of X,Y into DX,DY:

DX=abs(X-floor(X))
DY=abs(Y-floor(Y))

4. evaluate DX,DY to determine if painting white or black (1 or 0):

if (dx < .5 and dy < .5) or (dx > .5 and dy > .5)
      color=1
else
      color=0
endif

Because of the similarity of your images and mine, I just assumed our methods were equivalent in some hidden way, but now I think I was wrong.

Off course this doesn't include any antialias filter, and a lot of aliasing occur in the areas that the patterns become very small. The images I posted were generated with UltraFractal using the built-in antialias function. There's still some visible aliasing, tough.

My image has aliasing problems (mostly jagged lines) too, even though I applied a strong internal antialiasing filter. I don't use checkerboards in artistic pictures, mainly for this reason.

This is 5 iterations, with a small scale value for the pattern (1e-10)... take a look:
<Quoted Image Removed>

Interesting image. I love the sharp points.

[Fractal Ken]

Kali, one more thing . . . I really like your theory posts. They're understandable and thought-provoking.

[tomot]

very nice Kali!
can you do opposite colors also, to get a strobe effect?

[Kali]

@Ken: I really appreciate your comments 

@Tomot: Thanks! Not sure about the effect you mention, but see if this is what you are talking about...

(Looks like Mandelbrot on LSD  )

 

[tomot]

WOW! .......that got my attention, that's exactly what I'm talking about

[Kali]

I'm doing the same with Mandelbox 2D, sorry for starting the post into Mandelbrot category, perhaps the thread should be moved to another place (Anyway I will continue with Mandelbrot set later)

This is the -1.5 scale mbox sequence of the first 6 iterations  (first pic shows the plain grid, the last & larger one is after 6 iterations).
Is a good visualisation of how the formula works on the plane.

 
 
 



This is a close-up (10 iterations):

[tomot]

Very cool! please give me some background what our are using, and how you are implementing this?

willing and able to learn!

[Fractal Ken]

I'm doing the same with Mandelbox 2D, . . .

Mandelbox 2D sounds like the name of a program I would write.   Did you produce these images by implementing a 2D mandelbox formula in Ultra Fractal?

This is the -1.5 scale mbox sequence of the first 6 iterations  (first pic shows the plain grid, the last & larger one is after 6 iterations).
Is a good visualisation of how the formula works on the plane.

I use the 2D scale -1.5 box all the time; it's a very cool fractal.
I like how you've displayed the sequence of pictures by number of iterations. I'll have to remember this idea.

[Kali]

Very cool! please give me some background what our are using, and how you are implementing this?

willing and able to learn!

I'm using Ultrafractal 5 on this (answered @Ken question), but just for the laziness of implementing an antialias filter in VB.NET
You can download a trial version of UF5 if you don't have it, and I shall pass you the parameter file or just the code.
If you are a programmer tell me what language do you use and I can show you how to do it.

In general terms, the method is to iterate the formula you want, then take the final X and Y values, extract the fractional part into two variables (let's call it DX, DY), and paint the pixel with one of two different colors based on this condition:

if (DX < .5 and DY < .5) or (DX > .5 and DY > .5)
      pixelcolor=(1st color used)
else
      pixelcolor=(2nd color used)
endif

That's it!


@Ken: In fact, I looked forward the negative 1.5 scale 2D Mandelbox after seeing your excelent works you did with it... and yes, is a very cool fractal!
Now that I'm figuring out how it works, I wonder how Tglad discovered it... he has a really gifted mind, don't you think? I'm planning to find new ways of folding  based on his idea (I'm also beggining to think of mathematical operations as distortions & foldings of a grid)

[Fractal Ken]

I'm using Ultrafractal 5 on this (answered @Ken question), but just for the laziness of implementing an antialias filter in VB.NET

You can antialias without ever writing a filter. Just render a very large image, then scale it down using something like GIMP or Photoshop. Though I hate to admit it, this method works better than anything I've written.

In general terms, the method is to iterate the formula you want, then take the final X and Y values, extract the fractional part into two variables (let's call it DX, DY), and paint the pixel with one of two different colors based on this condition:

if (DX < .5 and DY < .5) or (DX > .5 and DY > .5)
      pixelcolor=(1st color used)
else
      pixelcolor=(2nd color used)
endif

This is a very clever idea.

Now that I'm figuring out how it works, I wonder how Tglad discovered it... he has a really gifted mind, don't you think?

YES

I'm planning to find new ways of folding  based on his idea (I'm also beggining to think of mathematical operations as distortions & foldings of a grid)

Here's a variation (for 2D) I've been exploring: Replace the boxfold with x = sin(PI*x), y = sin(PI*y).

[Vega]

I have the version too.

[superheal]

Reviving an old post!

What's the formula for the last two pictures, that vega posted.
Anyone knows?