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“Burning Ship” fractal

paulbourke.net

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Re: “Burning Ship” fractal

#21

Post by ogogmad »

ogogmad wrote: post #17

I've experimented with the "Mandelbar" (sometimes called the "tricorn") fractal over the split-complex numbers: 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".

Here it is. Split-complex burning ship using Xaos: 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

Re: “Burning Ship” fractal

#23

Post by goldenkey »

ogogmad wrote: post #18
goldenkey wrote: post #8

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…

If 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…

Complex 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_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 :-)

Re: “Burning Ship” fractal

#24

Post by ogogmad »

goldenkey wrote: post #23
ogogmad wrote: post #18

Earlier quoted context omitted.

If 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…

Complex 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_f... They're not someth…

A holomorphic function over the split-complex numbers is the direct sum of two holomorphic functions over R. In that sense, it's trivial.

You will be very disappointed once you see the split-complex Mandelbrot set. It will be an instructive exercise for you to see why it looks the way it does.

My point about complex conjugation is illustrated in [1]. Linear algebra over the complex numbers does not limit itself to the four arithmetic operations {+,-,*,/}, but uses complex conjugation as the 5th and final operation. Look at the definition of a unitary matrix, or a self-adjoint matrix, or a normal matrix. Also, look at the definitions of the QR decomposition and SVD. Complex conjugation is everywhere in linear algebra. Consequently, matrix theory over the split-complex numbers is actually somewhat interesting.

[1] - https://en.wikipedia.org/wiki/Matrix_decomposition

To some extent, this seems like a weird discussion about mathematical aesthetics.

Re: “Burning Ship” fractal

#27

Post by cmroanirgo »

BeetleB wrote: post #4

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.

Completely 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!

Re: “Burning Ship” fractal

#29

Post by MeteorMarc »

rjh29 wrote: post #15
rubatuga wrote: post #5

Great YouTube video of a deep zoom: https://www.youtube.com/watch?v=2S3lc2G3rWs

This 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.

How is it possible that most stages have symmetries around the center of view?

Re: “Burning Ship” fractal

#30

Post by wruza »

Oh, 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.

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