This 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.<p>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.<p>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.<p>[1] <a href="https://en.wikipedia.org/wiki/Algebraic_operation" rel="nofollow">https://en.wikipedia.org/wiki/Algebraic_operation</a><p>[2] <a href="https://en.wikipedia.org/wiki/Analytic_function" rel="nofollow">https://en.wikipedia.org/wiki/Analytic_function</a>
Xaos [1] is great for exploring fractals and the Burning Ship is no exception -- just define a user formula of<p><pre><code> powi(rabs(z)+i*rabs(im(z)),2)+c
</code></pre>
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:<p><pre><code> ;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)
</code></pre>
Note that this will also serve as a bit of stress test for your cpu ;-) (a rendered result is here: [2])<p>[1] <a href="https://github.com/xaos-project/XaoS" rel="nofollow">https://github.com/xaos-project/XaoS</a>
[2] <a href="https://i.imgur.com/mJ0uZG7.png" rel="nofollow">https://i.imgur.com/mJ0uZG7.png</a>
Great YouTube video of a deep zoom: <a href="https://www.youtube.com/watch?v=2S3lc2G3rWs" rel="nofollow">https://www.youtube.com/watch?v=2S3lc2G3rWs</a>
Glad 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.
I 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<p><a href="https://en.m.wikipedia.org/wiki/Burning_Ship_fractal" rel="nofollow">https://en.m.wikipedia.org/wiki/Burning_Ship_fractal</a>
Glad to see people showing some love to fractals !<p>A year ago, I built a tool to explore the Mandelbrot set fractal on the browser using vanilla JS.<p>Repo : <a href="https://github.com/yzdbg/mandelbrotExplorer" rel="nofollow">https://github.com/yzdbg/mandelbrotExplorer</a><p>Demo : <a href="https://yzdbg.github.io/mandelbrotExplorer/" rel="nofollow">https://yzdbg.github.io/mandelbrotExplorer/</a>
These 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...<p><a href="https://i.imgur.com/JRfLy6R.mp4" rel="nofollow">https://i.imgur.com/JRfLy6R.mp4</a>
Question: 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?
I 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.
This video is amazing. Its one of my favourites.
"Inside the burning ship": <a href="https://youtu.be/yaPTk-DqT1g" rel="nofollow">https://youtu.be/yaPTk-DqT1g</a>
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.
Burning 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)...
This reminds me deeply of the lighthouses at Burning Man in 2016: <a href="https://www.google.com/search?q=burning+man+lighthouses" rel="nofollow">https://www.google.com/search?q=burning+man+lighthouses</a>
I've experimented with the "Mandelbar" (sometimes called the "tricorn") fractal over the split-complex numbers: <a href="https://imgur.com/PkzY7B6" rel="nofollow">https://imgur.com/PkzY7B6</a><p>The result isn't completely trivial, but isn't particularly impressive either.<p>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".
Oh, fractals thread and I have a couple of questions. Are fractal “features” enumerable? Finite?<p>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?<p>Do fractals exist with an infinite number of features?<p>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.
I made a generative NFT based on this<p>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<p><a href="https://imgur.com/gallery/XQR8tNw" rel="nofollow">https://imgur.com/gallery/XQR8tNw</a>
This Burning Ship fractal was recreated in just 140 characters of javascript, see it working here (with source code): <a href="https://www.dwitter.net/d/25882" rel="nofollow">https://www.dwitter.net/d/25882</a>