But does the essence of the 2D Mandelbrot purely
Some said it couldn't be done - that there wasn't a true analogue to aĬomplex field in three dimensions (which is true), and so there couldīe no 3D Mandelbulb. Yoshiaki's excellent Quasi-fuchsian fractal (see right), but it turned out that even that doesn't have the variety of the Mandelbrot after zooming in. Were everywhere, but nothing quite as organic and rich as the originalĢD Mandelbrot. On October 13th, 2006, Marco Vernaglione put out the question and challenge to the world with this memorable document. Scoured everywhere to find signs of the 3D beast, but nothing turned up. I went to great lengths to explore the concept, including the utilization of various spherical coordinate systemsĪnd adjusting the rotation of each point's 'orbit' after every This looks great, but zooming in will not reveal the variety of style that the Mandelbrot has. For other 'hot spots', try here, and this one from the inside.Ĭreated by Dr. (found just before this article was published actually):įull size shown here. The best shot I could find was this view from the YZ plane Zooming in reveals some interesting detail, but nothing trulyįantastic. Perhaps we shouldĮxpect an 'apple core' shape with spheres surrounding the perimeter,Īnd further spheres surrounding those, similar to the way that circles Same as the first, except this time we try only multiplying angle phi by two, but not theta.Īlthough the second one looks somewhat impressive, and has theĪppearance of a 3D Mandelbulb very roughly, we would expect the realĭeal to have a level of detail far exceeding it. Also see Thomas Ludwig's globally illuminated render, and this one from Krzysztof Marczak. Multiplication bit (0.5*pi to theta and 1*pi to phi), to make it appearĪlmost 3D Mandelbrot-esque. One is the same as to the left, except offsets have been added to the Mandelbrot (zooming in doesn't show true 3D fractal detail). It's nice, but not exactly what I'd call a 3D But here's the somewhat disappointing result of theįormula (click any of the pictures for a larger view):įirst thing I tried was multiplying phi and theta by two, resulting in In theory, this could theoretically produce our amazingģD Mandelbrot. (complex multiplication), as in the normal 2D Mandelbrot, we rotateĪround phi and theta in 3 dimensional spherical coordinates ( see hereįor details). The basic idea is that instead of rotating around a circle
I then independently pictured the same concept and published the formula for the first time in November 2007 at the So the idea slumbered for 20 years until around 2007. The area a painstaking process to say the least. Require billions of calculations to see the results, making research in Rendering the 2D Mandelbrot, let alone the 3D version - which would Still can produce very interesting results - see later), and also wroteĪ short story about the 3D Mandelbrot in 1987 entitled " As Above, So Below" (also see his blog entry and notebook).īack then of course, the hardware was barely up to the task of
Mandelbulb (barring a small mistake in the formula, which nevertheless Around 20 years ago, along with otherĪpproaches, he first imagined the concept behind the potential 3D Ur story starts with a guy named Rudy Rucker,Īn American mathematician, computer scientist and science fictionĪuthor (and in fact one of the founders of the cyberpunk Opening Pandora's Box For the Second Time O