It looks a lot like a cubic grid with perspective projection; http://degreesgame.com/display/ShowImage?imageUrl=/storage/I...
A new kind of fractal?
21–30 of 53 posts
Re: A new kind of fractal?
#22It's not a fractal, but it is something familiar. Multiply two complex numbers z and c is equivalent to taking z and applying a rotation and dilation to it, the rotation through arg(c) and the dilation through |c|. Division is the inverse of both, so z/c is z rotated by -arg(c) and dilated by 1/|c|. What you're looking at, then, is taking the operation defined by c (rotate by -arg(c) and dilate by 1/|c|) and asking,…
I don't really get what you are describing. But perhaps you could provide a matlab script to show?
Re: A new kind of fractal?
#23It's not a fractal, but it is something familiar. Multiply two complex numbers z and c is equivalent to taking z and applying a rotation and dilation to it, the rotation through arg(c) and the dilation through |c|. Division is the inverse of both, so z/c is z rotated by -arg(c) and dilated by 1/|c|. What you're looking at, then, is taking the operation defined by c (rotate by -arg(c) and dilate by 1/|c|) and asking,…
> Now, a fractal is a set with a fractional Hausdorff dimension. Is there an authoritative definition of a fractal? The one you use rules out structures like Hilbert curves, which are generally considered fractals.
No, it's pretty vague. http://en.wikipedia.org/wiki/Fractal#Characteristics Also, the Hausdorff dimension of a Hilbert curve is 2.
Re: A new kind of fractal?
#24It's not a fractal, but it is something familiar. Multiply two complex numbers z and c is equivalent to taking z and applying a rotation and dilation to it, the rotation through arg(c) and the dilation through |c|. Division is the inverse of both, so z/c is z rotated by -arg(c) and dilated by 1/|c|. What you're looking at, then, is taking the operation defined by c (rotate by -arg(c) and dilate by 1/|c|) and asking,…
> Now, a fractal is a set with a fractional Hausdorff dimension. Is there an authoritative definition of a fractal? The one you use rules out structures like Hilbert curves, which are generally considered fractals.
> Is there an authoritative definition of a fractal?
Sort of. I believe Mandelbrot's original definition required only that the Hausdorff dimension exceeds the topological dimension. And that kinda-sorta includes Hilbert curves, if you count them as topologically 1-dimensional.
But I've seen other definitions, including a rather hazy one that was not a definition in the formal sense, but just talked about properties that certain interesting sets tend to have: self-similarity, etc.
In any case, I have yet to find a situation in which the formal definition of "fractal" actually mattered significantly. (If someone knows of one, I'd be interested.)
Re: A new kind of fractal?
#25From my sketchy understanding of your description of the algorithm, it seems like a 2d version of the patterns in gaps you see when you look at cornrows.
Re: A new kind of fractal?
#26Earlier quoted context omitted.
Ok, I have put a description of the algorithm above the comment section.
How do you calculate that percentage, do you brute force it or do you have a formula that estimates or exactly gives the percentage? Also, do you allow the result of the division to be near a gaussian integer, or do you do an exact floating point equals?
I brute force it up to n sample points. For the first image n was 100.
Yes, I allow the result to be near a gaussian integer. For the first image I counted everything as a gaussian integer where (real%1<0.1 && imaginary%1<0.1)
Re: A new kind of fractal?
#27It's possible that what we're looking at is based on the map f(z) = 1/z applied to the cartesian coordinate grid.
This is old-school, but a long time ago I wrote a java applet to help explore this function on the complex plane:
http://www.math.nyu.edu/~neylon/applet1/
If you move around the red square, you see a lot of the circular shapes that look similar to those in the fractal.
I don't completely understand the algorithm being used, so I'm not sure how to gain more confidence on this guess.
Re: A new kind of fractal?
#28It's not a fractal, but it is something familiar. Multiply two complex numbers z and c is equivalent to taking z and applying a rotation and dilation to it, the rotation through arg(c) and the dilation through |c|. Division is the inverse of both, so z/c is z rotated by -arg(c) and dilated by 1/|c|. What you're looking at, then, is taking the operation defined by c (rotate by -arg(c) and dilate by 1/|c|) and asking,…
madhadron: thank you for that explanation. In particular, I found your 1-dimensional analogy very helpful. Now that I understand the graph, I think it's even prettier. The beauty of these 'structural relationships' between numbers never ceases to amaze me.
Re: A new kind of fractal?
#29The self-similar structure originating from simple division looks reminiscent of various representations of the natural prime numbers.
That's what I thought of to.
Re: A new kind of fractal?
#30It would be a lot more useful if he also described how he generated the picture, which would enable mathematicians to give him some useful pointers for exploring it further.
Wow, HN front page! Ok, will describe the algorithm. One moment...
You have the 4000x4000 file with a jpg suffix but it is actually a png? I ran the image through pngcrush and it decreased the file size by 50%. You might want to do the same and save on bandwidth / page loading...
$ du -sh fractal*
14M fractal-optim.png
27M fractal.png