7 ms·
A new kind of fractal?
I stumbled across this and wonder what it is.
- crntaylor 14y agoIt 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.
- deleted 14y ago[deleted]
- no_gravity 14y agoWow, HN front page! Ok, will describe the algorithm. One moment...
- no_gravity 14y agoOk, I have put a description of the algorithm above the comment section.
- Nursie 14y agoCool, how high up the gaussian integers do you go to get that sort of detail?
- no_gravity 14y agoFor 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.
- rsiqueira 14y agoyes please, put the javascript source.
- Aardwolf 14y agoHow 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?
- no_gravity 14y agoI 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)
- dfc 14y agoYou 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
- no_gravity 14y agoYou are right! I had a png and a jpg version and uploaded the wrong one. Now the jpg is up. Thank you!
- sophiabatka464 14y agoBig Miracle 2012 Story Line:- Big Miracle 2012 is a very nice Drama and Romance movie.In small town Alaska, a news reporter recruits his ex-girlfriend – a Greenpeace volunteer – on a campaign to save a family of gray whales trapped by rapidly forming ice in the Arctic Circle.Inspired by the true story that captured the hearts of people across the world, the rescue adventure Big Miracle tells the amazing tale of a small town news reporter and a Greenpeace volunteer who are joined by rival world superpowers to save a family of majestic gray whales trapped by rapidly forming ice in the Arctic Circle. Local newsman Adam Carlson can’t wait to escape the northern tip of Alaska for a bigger market. watch now http://freemoviesite24.com/2012/10/big-miracle-2012-720p-hd-movie-free-download http://freemoviesite24.com/2012/10/big-miracle-2012-720p-hd-...
- gnosis 14y agoArchive: http://www.gibney.de.nyud.net/does_anybody_know_this_fractal http://www.gibney.de.nyud.net/does_anybody_know_this_fractal
- Sami_Lehtinen 14y agonyud.net doesn't really archive anything. https://en.wikipedia.org/wiki/Coral_cache https://en.wikipedia.org/wiki/Coral_cache
- papalalu 14y agoreminds me of points on a 3d grid.. http://www.youtube.com/watch?v=60CnwbngWWw&t=15m10s http://www.youtube.com/watch?v=60CnwbngWWw&t=15m10s
- mej10 14y agoThis looks like the complex differentiation of 1/z with a grid like domain.
- binarymax 14y agoVery cool. I stumbled upon something in the late 90s that is fractal in nature but doesnt use complex numbers - it just plots points directly on a grid with no need for iteration. http://www.youtube.com/watch?v=uwjhRYZ_eSI&feature=plcp http://www.youtube.com/watch?v=uwjhRYZ_eSI&feature=plcp
- EvilTerran 14y agoI'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-colour http://www.wikipaintings.org/en/m-c-escher/square-limit-colo...
- rouan 14y agoCan we have code please?
- nitrogen 14y agoThe self-similar structure originating from simple division looks reminiscent of various representations of the natural prime numbers.
- jlgreco 14y agoThat's what I thought of to. http://en.wikipedia.org/wiki/Ulam_spiral http://en.wikipedia.org/wiki/Ulam_spiral
- madhadron 14y agoIt'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.
- beagle3 14y ago> 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.
- 9mit3t2m9h9a 14y agoI would look into Lobachevsky geometry or inversions on the plane to describe this.
- geon 14y agoIt looks a lot like a cubic grid with perspective projection; http://degreesgame.com/display/ShowImage?imageUrl=/storage/IMG_0659.PNG?__SQUARESPACE_CACHEVERSION=1342745883110 http://degreesgame.com/display/ShowImage?imageUrl=/storage/I...
- bramcohen 14y agoFrom 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.
- deleted 14y ago[deleted]
- tylerneylon 14y agoIt'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/ 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.
- yoos 14y agoThe lines look like those on an inverted plane: http://xahlee.info/SpecialPlaneCurves_dir/Inversion_dir/inversion.html http://xahlee.info/SpecialPlaneCurves_dir/Inversion_dir/inve...
- VLM 14y agoAre you using a large fixed point system or floating point? I hope the fractal-ness isn't just floating point rounding writ large.
- no_gravity 14y agoIts Javascript numbers. As far as I know they are always 64bit floating points. Yes, its possible that floating point artefacts play a role in this.
- mikecane 14y agoI have nothing mathematical to add here. Just want to say when I voted for this item this morning I waited and waited for others to vote, then I had to leave the PC for several hours. Before that, I kept looking at that image and hoping others would vote for it and I'd see some Comments. Thanks for voting for it and all these Comments!
- dmboyd 14y agoMy attempt using matlab(octave) to generate a 100x100 version of the image (using his function description): https://gist.github.com/3903785 https://gist.github.com/3903785 Hadn't figured out if there is a way of vectorising it further. Would appreciate any tips if anyone has any.
- dmboyd 14y agoI've managed to hack it into a semi-vectorised implementation which works up to 700x700 before it runs out of memory. http://i.imgur.com/xzgXW.png http://i.imgur.com/xzgXW.png
- deleted 14y ago[deleted]
- no_gravity 14y agoInteresting! You dont have the squares. This is a hint that the squares in my version are due to floating point limitations.
- dmboyd 14y agoIt's probably a combination of floating point errors adding up.
- iazid 14y agoProbably you still can define a fractal by changing slightly the construction. Anyways, here's my take for a generalization : http://imgur.com/qD0aZ http://imgur.com/qD0aZ
- knicholes 14y agoTo me, it looks VERY similar to the Laue pattern of the enzyme Rubisco. http://www.amazon.com/Photographic-diffraction-Science-Photo-Library/dp/B007DJDN2S http://www.amazon.com/Photographic-diffraction-Science-Photo...