I'm reminded of Escher's "Square Limit" works. Not sure if this is actually the same class of grid as those he used to generate them, but it's visually similar at least:
http://www.wikipaintings.org/en/m-c-escher/square-limit-colo...
A new kind of fractal?
11–20 of 53 posts
Re: A new kind of fractal?
#12Can we have code please?
Re: A new kind of fractal?
#13Re: A new kind of fractal?
#14The self-similar structure originating from simple division looks reminiscent of various representations of the natural prime numbers.
Re: A new kind of fractal?
#15Earlier quoted context omitted.
Ok, I have put a description of the algorithm above the comment section.
Cool, how high up the gaussian integers do you go to get that sort of detail?
For the first image up to 100+100i. Please be aware, that I only use gaussian integers of the form x+xi where the real and imaginary part are the same. Im not sure if that is clear from my description. If there is demand, I can put up the source. Its Javascript.
Re: A new kind of fractal?
#16It'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, if you take the Gaussian integers as the vertexes of a directed graph, what fraction of the vertexes are the source of an edge.
Consider the 1 dimensional analogy using real numbers. Given some real number c, take all the integers as the vertexes of a graph, and if z/c (for some integer z) is also an integer, I put a directed edge from z to z/c. When are these connected? Well, if c is irrational, never. If c is rational, then there will be an infinite number of connections, but how infinite? When c is 2, there will be twice as many edges on average in any subset of the source vertexes as when c is 4. If we can write c as p/q, then the smaller p is, the more edges we'll get, and the brighter the pixel in your image.
The 1 dimensional analogy will have a spike at 1/2, smaller spikes at 1/3 and 2/3, yet smaller spikes at 1/4 and 3/4, smaller ones yet at 1/5, 2/5, 3/5, and 4/5, etc. The spikes will all be distinct (because between any two rationals there is an irrational), but will be infinitely close (because the rationals are dense in the reals). As you keep zooming in, you will get more and more edges like this.
What you're seeing is a variation on the classical structure of the rationals dense within the reals.
Now, a fractal is a set with a fractional Hausdorff dimension. We have to extract a set from your function of c in order to talk about fractal dimension. We could take the support of the function (everywhere it's not zero). In the one dimensional case, that's the rationals. We could take level sets farther up (the set of c such that f(c) = k, for a constant k). Those are subsets of the rationals. However, the rationals, while dense in the reals, are of measure zero in the reals, and have Hausdorff dimension zero, and so do all the level sets. So it's not a fractal.
Doesn't make it any less pretty though.
Re: A new kind of fractal?
#17I would look into Lobachevsky geometry or inversions on the plane to describe this.
Re: A new kind of fractal?
#18It'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,…
It is as fractal as this is: http://en.m.wikipedia.org/wiki/Hilbert_curve
Re: A new kind of fractal?
#19It'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.
Re: A new kind of fractal?
#20Earlier quoted context omitted.
Wow, HN front page! Ok, will describe the algorithm. One moment...
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?