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I don't think the author understands fractals well. I just skimmed the first part of the article and there are at least two really wrong ideas. The Mandelbrot
by danbruc 5y ago
I don't think the author understands fractals well. I just skimmed the first part of the article and there are at least two really wrong ideas.
The Mandelbrot set is a set but the images are not generated by evaluating some function like a sine function that traces out the border of that set and then coloring the exterior with nice colors. Instead the colors represent the escape times of points and the trajectories of those obtained by iterated evaluation of the defining function are not actually visible. Buddhabrot [1] visualizes the trajectories to some extend but more for artistic beauty than clear visualization.
Secondly there is no third dimension that you access by zooming such that each image is just a slice through some three dimensional object. Yes, one can do this and look at the Mandelbrot set as a slice through a higher dimensional object but zooming into the Mandelbrot set in your standard fractal drawing app really just magnifies parts of it. Any three dimensional appearance is just an [intended] artifact of the color palette used.
EDIT: Maybe the app used treats the escape time as height coordinate turning the Mandelbrot set into a three dimensional landscape and that is what he is trying to say. While a common idea for visualizations it is not an intrinsic aspect of the Mandelbrot set.
EDIT: I just realized that the author also wrote the fractal visualizer used to make those images. Maybe I am just misunderstanding his words or he is not great at explaining what he means or he wrote the app without a good understanding what it actually does...I don't know. Especially given that he has seemingly worked in computer graphics for a long time according to Wikipedia [2].
[1] https://en.wikipedia.org/wiki/Buddhabrot https://en.wikipedia.org/wiki/Buddhabrot
[2] https://en.wikipedia.org/wiki/Kai_Krause https://en.wikipedia.org/wiki/Kai_Krause
- crazygringo 5y agoBelieve me, he definitely understands fractals well. :) But I was also thrown off by the sentence about a "2D slice of a 3D zoomable construct". I think it's just sloppy writing, and indeed he's describing the escape times as generating a 3D surface that you can zoom "deeper" into with more iterations. Still not really a 2D slice... but given his creations I'll just assume maybe he hadn't had his morning cup of coffee yet... I remember using Kai Power Tools back in the 90's! Certainly one of the most creative interfaces at the time as well...
- aardvark179 5y agoThere are quite a few ways to map aspects of points outside the set to colours or other variables. It’s been a long time since I looked at Frax but I seem to remember it maps iterations, the direction in which points are escaping, and a few other things to a height, colour index, speculative index, etc., and then renders the set by rendering that surface. It’s not as fun as the four dimensional construct that is all the Julia sets, but probably easier to explain to people.
- ryanmentor 5y agoNot three dimensions, but fractional and fractal dimensions : https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_...
- danbruc 5y agoThis are different things. The Mandelbrot set iterates z <- z² + c usually with z initialized to zero and c varying across the complex plane with each value corresponding to a pixel in the image. But you can also vary the initial value of z and then the standard Mandelbrot set is a two dimensional slice through a four dimensional generalized Mandelbrot set. You can add another degree of freedom by varying the exponent. The Hausdorff dimension on the other hand characterizes the fractal structure itself, not the number of degrees of freedom the fractal is defined over.