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< title > Fractal galery< / title >
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< h1 > 2D and 3D fractal renders< / h1 >
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< h1 > The Results< / h1 >
We all like fancy images, so here are some results: (the slideshow is draggable as well)
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< button class = "slider-control slider-fullscreen" > Fullscreen< / button >
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< h1 > Can you animate the Mandelbulb?< / h1 >
< p > Yes, of course! It only takes a huge amount of time. This one rendered in about an hour, it animates the power from 1 to 20:< / p >
< video src = "https://datenvorr.at/renders/mandelbulb.mp4" controls = "show" > < / video >
< h1 > The Code< / h1 >
< i > coming soon< / i >
< h1 > What is a Mandelbulb?< / h1 >
< p > A mandelbulb is a 3D Fractal, which was formulated in 2009 by Paul Nylander. It takes the known approach of \(z_{n+1} \to z_n^2+c\).
But he defines the < i > nth power< / i > of a vector \(z\) (\(z^n\)) a little more complex:< / p >
< p > Let \(v = (x,y,z)\) be a vector in \(\mathbb{R}\), then \(v^n := r^2 \cdot (\sin(n\theta)\cos(n\phi), sin(n\theta)sin(n\phi),cos(n\theta))\) where
\[
r = |x| \\
\phi = \arctan(y/x) \\
\theta = \arctan\left(\frac{\sqrt{x^2+y^2}}{z}\right)
\]
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< p > The 3D Mandelbulb is now defined as the set of points, where the orbit of \(z_n\) is bounded (the vector does not grow indefinitely).< / p >
< p > This is where the < i > power< / i > comes in (mentioned in the image descriptions). We can look at Mandelbulbs with powers that are not 2.< / p >
< h2 > What happens when the power is & lt 1?< / h2 >
< p > Short answer: Weird stuff. Long answer: I don't know!< / p >
< p > here are some renders with weird powers:< / p >
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< img class = "embed-img-small" src = "https://datenvorr.at/renders/lowres/mandelbulb0.2.png" alt = "A mandelbulb rendered with power 0.2, it's just a small sphere" / >
< span class = "subtitle" > Mandelbulb with power 0.2< / span >
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< img class = "embed-img-small" src = "https://datenvorr.at/renders/lowres/mandelbulb-2.png" alt = "A mandelbulb rendered with power -2, it's super weird looking" / >
< span class = "subtitle" > Mandelbulb with power -2. Looks like a giant, interdimensional jellyfish.< / span >
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< b > Source code:< / b >
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< li > < a href = "https://git.datenvorr.at/anton/fractals2d.h" target = "_blank" > 2D Mandelbrot rendering< / a > < / li >
< li > < a href = "https://git.datenvorr.at/anton/raymarcher.h" target = "_blank" > 3D Fractal rendering< / a > < / li >
< li > < a href = "https://git.datenvorr.at/anton/images.h" target = "_blank" > Image library and bitmap encoder< / a > < / li >
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< b > Legal stuff:< / b > < br / >
© Copyright by Anton Lydike < br / >
All fractal images and videos are hereby made public domain< br / >
Image carousel built with < a href = "https://github.com/pawelgrzybek/siema" target = "_blank" rel = "noopener" > siema< / a >
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