OK. Here's my last attempt at trying to explain why this isn't a fractal. This is an image similar to what you posted:

This is what happened after I zoomed in a bit and I used anti-aliasing:

Started by fir, Feb 05 2014 11:23 AM

47 replies to this topic

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Posted 07 February 2014 - 07:55 AM

for me i may repeat it seem this is not worse fractal than sierpiński carpet

I have no idea why you still think your image is a fractal. The way I see it, what you plotted is a couple of hundred concentric circles, which when sampled with a regular grid result in a spectacular moiré pattern. It's not like we are saying your image isn't pretty: It just has little to do with fractals.

I found this link: http://www.nahee.com/spanky/www/fractint/circle_type.html (Notice the "not a fractal" part.)

I think it is unrelated to sampling on the grid- all in all this is well defined

F(x,y) function for x,y are real, - so sampling to a grid is not important imo, it just blurs the details, dont you think?

Very good info in this link, (i was searching for such references) though here is written ". The resulting image is not a fractal because all detail is lost after zooming in too far. "

Im not sure if this is true, if one will raise the frequenzy of palette

i think the detail depth will probably increase to infinity - so it probably depends how you define this construct

Your function F(x,y) is a continuous function, but when you create your image you sample F(x,y) at discrete points. Each pixel in the image is a sample point of the function. In your first post, for example, you sample F(x,y) at roughly 950 discrete points along both the X and the Y axis. The interference is not in F(x,y) itself, but comes from sampling it at discrete points.

Posted 07 February 2014 - 08:23 AM

You got to low frequenzy paltette (*)- I was saing about this, (more than once i think and I see some ignore it ;\ - ) the details will appear if you increase it - if you will ignore this we will not agree here

(same thing with sierpiński on mandelbrot if you do only 5 iteration steps

you will get finite complexity)

(*) and maybe to low sampling frequenzy too, if this pattern vanishes

indeed maybe the grid sampling is needed - i dont know if this distortion to circles are so small that this grid sampling shows it

anyway this is strange - do antyaliasing destroyed most of the pattern but leaved some horizontal and vertical line ones? if the rest vanished why the vertical horizontal are still visible?

what way this antyaliasing works here? average of many subsamples per pixel?

on the other way this not changes too much:

if underlying shape is such smooth, indeed the grid sampling should be included in algorith - but this do not change to much only adds some module to formula - sample the result with rectangle grid

you think infinite complexity is not obtainable this way?

**Edited by fir, 07 February 2014 - 09:25 AM.**

Posted 07 February 2014 - 08:26 AM

Your function F(x,y) is a continuous function, but when you create your image you sample F(x,y) at discrete points. Each pixel in the image is a sample point of the function. In your first post, for example, you sample F(x,y) at roughly 950 discrete points along both the X and the Y axis. The interference is not in F(x,y) itself, but comes from sampling it at discrete points.

Dont think so, Could you explain this, lets say that you are taking

some point x, y = 0.1776527, 0.23876 You say that color value of this point depends on the grid resolution? IMO F(x,y) values are grid independant

Posted 07 February 2014 - 08:50 AM

Interference does not happen in one single sample point. It is the interaction between neighboring sample points. You cannot say anything about interference in one point. It is irrelevant what the value is in one point of the function.Your function F(x,y) is a continuous function, but when you create your image you sample F(x,y) at discrete points. Each pixel in the image is a sample point of the function. In your first post, for example, you sample F(x,y) at roughly 950 discrete points along both the X and the Y axis. The interference is not in F(x,y) itself, but comes from sampling it at discrete points.

Dont think so, Could you explain this, lets say that you are taking

some point x, y = 0.1776527, 0.23876 You say that color value of this point depends on the grid resolution? IMO F(x,y) values are grid independant

Posted 07 February 2014 - 03:36 PM

Another proof it's a sampling artifact: Zoom in and out on Álvaro's anti-aliased image with your browser (IE,Firefox, Opera: CTRL + mouse wheel) and watch these circles come and go. One can sometimes even notice a second Moiré effect in a checkerboard fashion probably due to a box filter for the images.

Posted 07 February 2014 - 08:52 PM

Another example of moire pattern. If you have an LCD monitor, scroll slowly across Alvaro's second image above. You get an interference pattern as well, and the scrolling speed changes the frequency of the fringes. I don't know if this will work with a CRT monitor since they use electron beams instead of a discrete crystal lattice.

The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

- *Pessimal Algorithms and Simplexity Analysis*