Pretty 3D fractals were everywhere, but nothing quite as organic and rich as the original 2D Mandelbrot.
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On October 13th, 2006, Marco Vernaglione put out the question and challenge to the world with this memorable document. I 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 systems and adjusting the rotation of each point's 'orbit' after every half-turn of phi or theta. 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. The best shot I could find was this view from the YZ plane (found just before this article was published actually):įull size shown here. Zooming in reveals some interesting detail, but nothing truly fantastic. Perhaps we should expect an 'apple core' shape with spheres surrounding the perimeter, and further spheres surrounding those, similar to the way that circles surround circles in the 2D Mandelbrot. 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 the appearance of a 3D Mandelbulb very roughly, we would expect the real deal to have a level of detail far exceeding it. Also see Thomas Ludwig's globally illuminated render, and this one from Krzysztof Marczak. This one is the same as to the left, except offsets have been added to the multiplication bit (0.5*pi to theta and 1*pi to phi), to make it appear almost 3D Mandelbrot-esque. It's nice, but not exactly what I'd call a 3D Mandelbrot (zooming in doesn't show true 3D fractal detail). The first thing I tried was multiplying phi and theta by two, resulting in the shape you see above. But here's the somewhat disappointing result of the formula (click any of the pictures for a larger view): In theory, this could theoretically produce our amazing 3D Mandelbrot. The basic idea is that instead of rotating around a circle (complex multiplication), as in the normal 2D Mandelbrot, we rotate around phi and theta in 3 dimensional spherical coordinates ( see here for details). I then independently pictured the same concept and published the formula for the first time in November 2007 at the web site. So the idea slumbered for 20 years until around 2007. Back then of course, the hardware was barely up to the task of rendering the 2D Mandelbrot, let alone the 3D version - which would require billions of calculations to see the results, making research in the area a painstaking process to say the least. Around 20 years ago, along with other approaches, he first imagined the concept behind the potential 3D Mandelbulb (barring a small mistake in the formula, which nevertheless still can produce very interesting results - see later), and also wrote a short story about the 3D Mandelbrot in 1987 entitled " As Above, So Below" (also see his blog entry and notebook). Ur story starts with a guy named Rudy Rucker, an American mathematician, computer scientist and science fiction author (and in fact one of the founders of the cyberpunk science-fiction movement).
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Opening Pandora's Box For the Second Time O