Fractal patterns from decimal extraction

[Kali]

I suppose this is already known, but I discover it by myself:
If you take a grid of points from a 2d plane (just like you do to evaluate a set of points in a escape-time fractal), and then, without using any formula at all, just extracting the decimals from each x and y value to evaluate them, you get a fractal pattern.

e.g. for each point (x,y), iterate:

p=p+1 // the decimal place to extract

and this is a method for extracting a decimal at position 'p'
and making it a positive 1 digit integer:

xx=abs(floor(x*(10^p))-floor(x*(10^p)/10)*10)
yy=abs(floor(y*(10^p))-floor(y*(10^p)/10)*10)

(use floor() or int() or whatever that cuts out the decimal part)

then set up a bailout condition like this:

xx+yy>10

and coloring based on escape time, you get this Sierpinski-like pattern:



Maybe just a curiosity, but I wanted to share it with you, folks.

[Fractal Ken]

What a cool and creative idea!

[Kali]

Thanks, Ken

This one is made using only absolute values of x,y and iterating z=z+xx-yy

[Jameses]

any known correspondind UF forumlas?

[Kali]

any known correspondind UF forumlas?

This is an example code for UF:

decimalextraction {
init:
float x=real(#pixel)
float y=imag(#pixel)
int p=0

loop:
p=p+1
xx=abs(trunc(x*(10^p))-trunc(x*(10^p)/10)*10)
yy=abs(trunc(y*(10^p))-trunc(y*(10^p)/10)*10)
z=xx+yy ; (or z=z+xx-yy for the second image I posted)

bailout:
|z|<100
}

(Use multi-pass linear drawing method and turn off periodicity checking)

[Kali]

For the next one I used the fractional part from the modulus of x,y (sqrt(x*x+y*y)) and then I colored it based on the decimal digit at 8th place.
(The borders were darkened by me for a better visual impact)

[Kali]

A different coloring method, and with scaled up pixels (that means I lowered the resolution), to see the patterns better.



To change the resolution of the x-y values in UF, I used:

newx=trunc(x*res)/res
newy=trunc(y*res)/res

For higher values of 'res', more resolution.
The patterns also change when you change the resolution, because new values have different fractional parts.
I'll make a simple movie to show this.

[matsoljare]

You might get a better result if you use a lower base, such as 2 or 3.

[Kali]

Do you mean in the "10^pos" part?

One thing I'm not sure is if I'm calling this in the proper way. I mean "decimal extraction", because in spanish we refer to the digits of the fractional part of a number as "decimals". Perhaps in english I must say that I'm extracting the "fractional digits"?

[Kali]

This is a short movie of the changing patterns... note the variety and unique design of each one.

I think there's an important concept behind this, but I don't know yet what it is, I'm just following my intuition.
I have some ideas but I must research further before posting anything.

I didn't want a boring silent video, so the music is from my friends of Unsocial THC Syrah, part of their song "No Domino" (great song)

<a href="http://www.youtube.com/v/EGfgg9hFm2E?fs=1&amp;amp;hl=es_ES&amp;amp;rel=0&quot; type=&quot;application/x-shockwave-flash&quot; allowscriptaccess=&quot;always&quot; allowfullscreen=&quot;true&quot; width=&quot;800&quot; height=&quot;600&quot" target="_blank">http://www.youtube.com/v/EGfgg9hFm2E?fs=1&amp;amp;hl=es_ES&amp;amp;rel=0&quot; type=&quot;application/x-shockwave-flash&quot; allowscriptaccess=&quot;always&quot; allowfullscreen=&quot;true&quot; width=&quot;800&quot; height=&quot;600&quot;</a>

[Fractal Ken]

Kali, cool movie!

One thing I'm not sure is if I'm calling this in the proper way. I mean "decimal extraction", because in spanish we refer to the digits of the fractional part of a number as "decimals". Perhaps in english I must say that I'm extracting the "fractional digits"?

I think decimal extraction is a good name as long as you're using base 10.

[Edit: Removed an ill-conceived comment about aliasing.]

Regards,
Ken

[Kali]

Ken,
I was about to answer to your comment about aliasing, as I'm using the forum's setting to receive the replies on my email.
As for your edit, seems that you got it... The last patterns I posted (not the first ones), are in fact, sort of aliasing patterns of concentric circles that becomes thinner as the decimal position to be evaluated moves to the right. In a *perfect* ideal rendering system (I mean infinite resolution), the patterns won't appear no matter what decimal position is chosen, just concentric circles with different spacings between them.
So the patterns are defined by the resolution in the scanning for x-y values of the plane, and the decimal position evaluated.
There's still something in all this that have me thinking...

Thank you for your comments

[Fractal Ken]

Kali, what you've written makes sense. When I saw your movie, I was reminded of aliasing artifacts, which are usually undesirable. However, aliasing seems to be the source of your interesting patterns in this case.

[Kali]

This is one output from the software I use (wihout using the resolution lowering method)



And this is the same with a 1E6x zoom:



Aren't this called "moire patterns"?

[Fractal Ken]

Nice example, Kali. I believe your picture does illustrate a moiré pattern (a type of aliasing).