7 ms·
“Burning Ship” fractal
- ogogmad 4y agoQuestion: Do fractals have projective duals? In the projective plane RP^2, the dual of a point is a line, and vice versa. Do fractals like "Burning Ship" and "Mandelbrot" have projective duals, where each point in the set is replaced with a line? What would these look like?
- ronald_raygun 4y agoMmm. I don't think that's a super useful way to think about it. The way I've always conceptualized it was that if you look at fractals, they are determined by some complex polynomial. So you could look at the space of polynomials, and each point in there , would be its own fractal. For the right space of polys thats a Hilbert space, and so by Riesz representation theorem it would have a dual space, but I dont think thats super useful here
- Jaxan 4y agoI think Julia sets are perhaps what you are looking for.
- rsiqueira 4y agoThis Burning Ship fractal was recreated in just 140 characters of javascript, see it working here (with source code): https://www.dwitter.net/d/25882 https://www.dwitter.net/d/25882
- rep_movsd 4y agoI made a generative NFT based on this I manually searched the space by zooming and panning into various areas from 2x all the way to about 10^13 zoom ans rendered them in high intensity color schemes https://imgur.com/gallery/XQR8tNw https://imgur.com/gallery/XQR8tNw
- deleted 4y ago[deleted]
- isoprophlex 4y agoI really like the B/W images on this page! Very nice, somehow the greyscale renders are able to capture the mood a lot better than the traditional colored mappings on eg. Wikipedia https://en.m.wikipedia.org/wiki/Burning_Ship_fractal https://en.m.wikipedia.org/wiki/Burning_Ship_fractal
- BeetleB 4y agoGlad to see Paul Bourke is still active. I encountered his site almost 20 years ago and spent many hours on it. Fantastic stuff. Glad to see he still maintains a minimalist site.
- cmroanirgo 4y agoCompletely agree. His public work helped me with some 3d game development back in the mid 90's, mainly with geometry and 3d projections. Back then, in the pre-google era there wasn't a lot of places on the web to get good information, but Paul Bourke was there. He & his sites seem timeless somehow. Great to see it on HN!
- rubatuga 4y agoGreat YouTube video of a deep zoom: https://www.youtube.com/watch?v=2S3lc2G3rWs https://www.youtube.com/watch?v=2S3lc2G3rWs
- rjh29 4y agoThis is crazy. How do they pilot the zooming of the fractal? It must be carefully planned and programmed in to generate the video. edit: I guess if they find an interesting thing very zoomed-in, then zoom out from that, the whole video will be interesting.
- dmurray 4y agoI'm not sure your edit is right. No reason why there can't be a tiny but very complex thing which looks single-colour at the next 1 or 10 or 100 orders of magnitude. But there are an infinite number of very interesting things in this fractal. If you constantly zoom in on interesting looking areas, you will find this kind of complexity with minimal need for backtracking.
- mike_hock 4y agoI'm guessing most fractals don't have a smooth boundary anywhere, so if you know any boundary point, you can zoom in on it and it'll be "interesting."
- danwills 4y agoYeah I think you're right mike_hock. On top of that I think that locations on (or near) the boundary, tend to stay on the boundary (and stay in the center of the image too) when zooming out. While purely zooming (not translating) to a known boundary (or near) point, you won't ever see a move to another section-of-boundary, so if there are both 'inside' and 'outside' regions, corresponding to attractors at 0 and infinity, (the 2 main ones in these types of fractals) in the most-zoomed in state, then there will always be regions of both states contained in the final image when zoomed out (until you get to the 'top'). Maybe it would be possible for there to be formulae that don't hold to this? If the fractal had an incredibly sparse structure, say? To be honest I'm more interested in the opposite myself: Structures where the boundary (between N regions or behaviors) is so wiggly, it's almost 2 dimensional itself!. (If anyone wants to read more, I've called one particular interesting example of this: 'mandelfield' on UltraIterator)
- Tao3300 4y agoI used to spend a ton of time in the Xaos program poking around the Burning Ship. You could middle-click and it would make a Julia set based on the point you clicked, and you could find some amazing stuff in the rigging of the ship.
- goldenkey 4y agoThis fractal is non-holomorphic which makes it kind of a dumpster fire..or a burning ship I guess... The easiest layman's definition of holomorphic is any function or operations on complex numbers that doesn't peek at the real and imaginary parts of the complex number. In algebra, there is no combination of algebraic operations [1] that can do so. But on computers, we have Re(z) and Im(z), which are easy to calculate since the complex number is simply stored as two parts, and it just pulls one of them out. When you violate this sacred notion, of treating a complex number as a singular entity, indivisible, the operations or functions you get, no longer have nice properties, like being analytic [2]. You are no longer doing algebra, you have gone into the realm of chaotics over R^2 with an equivalence relation through rotations, which is a far different beast. That's why other fractals that treat complex numbers as first class citizens, singular entities, and don't use hacks like Re or Im, are much prettier and have less branch cuts. [1] https://en.wikipedia.org/wiki/Algebraic_operation https://en.wikipedia.org/wiki/Algebraic_operation [2] https://en.wikipedia.org/wiki/Analytic_function https://en.wikipedia.org/wiki/Analytic_function
- ogogmad 4y agoIf you replace the complex numbers with split-complex numbers, then the set of holomorphic functions becomes uninteresting. You therefore need to use * (meaning the split-complex analogue of complex conjugation) to get anything interesting. It follows that the obvious analogue of complex analysis over the split-complex numbers is pretty sterile, but I think that the split-complex numbers themselves remain interesting. The operation * (meaning conjugation) isn't totally ugly. The reason for that is because it shows up in matrix theory. A lot of families of matrices (like the unitary and the self-adjoint matrices) are defined using *. Therefore, while the function theory of the split-complex numbers might be boring, their matrix theory is still somewhat interesting because of its dependence on *. The relevance to generating fractals using the split-complex numbers is that you need to use non-holomorphic functions to get interesting results. Some results here: https://news.ycombinator.com/item?id=32211495 https://news.ycombinator.com/item?id=32211495
- goldenkey 4y agoComplex conjugation is non-holomorphic and yes, it's ugly. I still don't see why it being useful as a construction makes it a beautiful operation in analysis given that it isn't algebra. There are holomorphic split-complex functions and they aren't uninteresting at all - they are quite beautiful and complex, relating to wave equations: https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_functions https://en.wikipedia.org/wiki/Motor_variable#D-holomorphic_f... They're not something often looked into in the fractal scene but if anyone has spare time, I'd love to see a D-holomorphic fractal analog of the Mandelbrot :-)
- dang 4y agoRelated: Burning Ship Fractal - https://news.ycombinator.com/item?id=12581569 https://news.ycombinator.com/item?id=12581569 - Sept 2016 (65 comments)
- dr_kiszonka 4y agoJust curious - do you have a script that finds related posts for each submission or do you rely on your memory, Dang?
- layer8 4y agohttps://hn.algolia.com/?query=burning%20ship%20fractal&sort=byDate&type=story https://hn.algolia.com/?query=burning%20ship%20fractal&sort=...
- dang 4y agoKind of a script, but not quite. Here's a pointer to past explanations: https://news.ycombinator.com/item?id=29370676 https://news.ycombinator.com/item?id=29370676. Edit: or, to dereference all the way: https://news.ycombinator.com/item?id=26158300 https://news.ycombinator.com/item?id=26158300.
- jimmygrapes 4y agoIf it helps any, here's the post you made that describes it most simply without needing to go 5 or 6 levels deep of self reference to the same question: https://news.ycombinator.com/item?id=26156901 https://news.ycombinator.com/item?id=26156901 (I had to click a lot of "here's a post where I explained it before" links that had "here's a post where I explained it before" before I found this one)
- pvg 4y agoI think you mean https://news.ycombinator.com/item?id=26158300 https://news.ycombinator.com/item?id=26158300 as Umberto Eco's been dead since 2016 and probably did not moderate HN while alive.
- anewpersonality 4y agoIncredibly creepy. What if one day we found a fractal that contains a message from God?
- 867-5309 4y agothis one's message seems pro-piratical
- pavlov 4y agoIt would be like finding the face of Jesus on a piece of toast, or the face of an alien on a satellite image of Martian surface. (Both of which were actual images that gathered some cult following in their time.) Spend enough time applying human interpretation to non-human processes, and something will eventually come up that tickles your senses.
- hgazx 4y agoWhen I was a kid I had a sound editing application (think of something like audacity) and it had a tool to generate white noise. I was deathly afraid of generating white noise and then slowing it down because I thought that I may find hidden messages.
- can16358p 4y agoGenerate enough digits of pi and you'll find it. Also check Library of Babel, there are some there too. /s
- forinti 4y agoI play around with fractals a lot and once I fiddled with mandelbrot and found something that resembled a pig running on two legs. I guess it would be called the running pig fractal. This might be a message from god: don't eat pork.
- ronald_raygun 4y agoI think we already found that ;) https://en.wikipedia.org/wiki/Buddhabrot https://en.wikipedia.org/wiki/Buddhabrot
- m12k 4y agoGreat, now I want to play Return of the Obra Dinn again
- ogogmad 4y agoI've experimented with the "Mandelbar" (sometimes called the "tricorn") fractal over the split-complex numbers: https://imgur.com/PkzY7B6 https://imgur.com/PkzY7B6 The result isn't completely trivial, but isn't particularly impressive either. Anybody want to try Burning Ship over the split-complex numbers? It looks like you only need to replace the complex "i" with the split-complex "j".
- ogogmad 4y agoHere it is. Split-complex burning ship using Xaos: https://imgur.com/a/tBTuUDV https://imgur.com/a/tBTuUDV This is my first time using that program. It doesn't look too bad. Same thing, but with the dual numbers: https://imgur.com/a/kTU5ztn https://imgur.com/a/kTU5ztn
- ogogmad 4y agoA follow-up question, not easily answered using Xaos: The dual numbers can be pictured as lines in the Euclidean plane instead of points (details here: https://en.wikipedia.org/wiki/Laguerre_transformations https://en.wikipedia.org/wiki/Laguerre_transformations). What would the above "dual number burning ship" look like?
- pantulis 4y agoI miss FRACTINT.
- yzdbgd 4y agoGlad to see people showing some love to fractals ! A year ago, I built a tool to explore the Mandelbrot set fractal on the browser using vanilla JS. Repo : https://github.com/yzdbg/mandelbrotExplorer https://github.com/yzdbg/mandelbrotExplorer Demo : https://yzdbg.github.io/mandelbrotExplorer/ https://yzdbg.github.io/mandelbrotExplorer/
- wruza 4y agoOh, fractals thread and I have a couple of questions. Are fractal “features” enumerable? Finite? By feature I mean in fractal zoom videos they pick some zoom path and it generates repeating shapes again and again until they switch to different path. How many repeating patterns there are? Do fractals exist with an infinite number of features? Do fractals exist where features cannot repeat in future zoom levels? Or at least that you barely could predict where a repeating part could be. Sort of a chaotic fractal.
- ronald_raygun 4y agoOh they fractal features are definitely infinite. I worked on a genetic algo a to breed fractals together a number of years ago, and one thing I noticed was that the highest non-zero poly power would determine the shape of the fractal - so a c^3 fractal would have "three petals", a c^4 would have "four petals". So clearly these things can have infinite features https://github.com/zoso95/genetic-algorithm-fractals https://github.com/zoso95/genetic-algorithm-fractals
- kenshoen 4y agoThese kind of fractal have actually 4 dimentional structure since c_x and c_y can also be parameters. I'd love to see their slice as 3D, but have yet to find a good way to visualize them... https://i.imgur.com/JRfLy6R.mp4 https://i.imgur.com/JRfLy6R.mp4
- messe 4y ago> These kind of fractal have actually 4 dimentional structure since c_x and c_y can also be parameters They're not parameters in that sense. The fractal is computed by taking each point on the plane as coordinates (c_x, c_y), and then iteratively applying the recursion relation. Then, with luminosity depending on how quickly that sequence escapes to infinity, we color in that point (c_x, c_y) in our image.
- ithinkso 4y agoThat's not what kenshoen meant. For example for the Mandelbrot set we have a function f(z) = z^2 + c for complex z,c and each pixel in the image represents c_x,c_y in c = c_x + i*c_y and then you iterate f(0), f(f(0))... On the other hand, if you have c constant and each pixel represents z_x, z_y in z=z_x + i*z_y instead, then iterating f(z),f(f(z))... gives you a Julia set. But you can think of f as a function of two complex arguments f(z,c)=z^2 + c and iterate it on the whole domain (two complex = four real dimensions) and then have a picture being a slice through any 2D or (even 3D, which is what parent is talking about) plane you like. In other words, the famous Mandelbrot fractal picture is a slice of f(z,c) through a plane z=0, and Julia set pictures are slices through planes c=constant but there is no reason one cannot make other pictures of f(z,c) (just be careful what you meant by iterating a function f: C^2 -> C). The burning ship fractal in the article is the same but the function f(z,c) is a bit weirder
- rngname22 4y agoVisions of Chaos perhaps?
- IngoBlechschmid 4y agoHere you can interactively explore the 4d mother of the Mandelbrot fractal and all its Julia fractals: https://rawgit.com/MatthiasHu/FractalsWebGL/4d/page.html https://rawgit.com/MatthiasHu/FractalsWebGL/4d/page.html
- azalemeth 4y agoXaos [1] is great for exploring fractals and the Burning Ship is no exception -- just define a user formula of powi(rabs(z)+i*rabs(im(z)),2)+c and set c to zero (or don't!). The views in the article are mirrored compared to xaos and are in the left area close to the axis. Some good views and palettes to explore interactively are (my taste!) around: ;Position file automatically generated by XaoS 4.2.1 ; - a realtime interactive fractal zoomer ;Use xaos -loadpos <filename> to display it (initstate) (filter 'anti #t) (palette 2 73629707 0) (formula 'user) (usrform "powi(rabs(z)+i*rabs(im(z)),2)+c") (usrformInit "0") (maxiter 5000) (bailout 5) (view -1.6924 -0.02769 0.1137 0.1137) Note that this will also serve as a bit of stress test for your cpu ;-) (a rendered result is here: [2]) [1] https://github.com/xaos-project/XaoS https://github.com/xaos-project/XaoS [2] https://i.imgur.com/mJ0uZG7.png https://i.imgur.com/mJ0uZG7.png
- akkartik 4y agoIs there a list of awesome fractals to zoom into with Xaos somewhere? edit: Wait a minute, it's right on the same site! http://www.paulbourke.net/fractals http://www.paulbourke.net/fractals
- davedx 4y agoIt’s the Staberinde from Use of Weapons.
- vcdimension 4y agoThis video is amazing. Its one of my favourites. "Inside the burning ship": https://youtu.be/yaPTk-DqT1g https://youtu.be/yaPTk-DqT1g You can see some thought has gone into not just the mathematics and rendering process, but also the artistic direction (lighting, choice of flight path) and music.
- rob74 4y agoBurning ship? Personally it reminds me more of a pair of slightly twisted gothic style church towers (Notre Dame de Paris anyone?), or the more slender version, of some kind of fantastic gothic style bridge towers (think Golden Gate Bridge)...
- tibbon 4y agoThis reminds me deeply of the lighthouses at Burning Man in 2016: https://www.google.com/search?q=burning+man+lighthouses https://www.google.com/search?q=burning+man+lighthouses