The idea actually entices me because I've always loved mathematics and problem solving. I tried using this as a starting point to further discover how it all worked, and failed: So when it's combined with concepts like eternity and chaos, I REALLY get interested, but for now and for the foreseeable future, I just don't have the time.īut, for anyone who cares to try on their own, here's the Mandelbulb formula from Daniel White's site (One of the original discoverers of the 3D MAndelbulb). Similar to the original 2D Mandelbrot, the 3D formula is defined by: What's the formula of this thing?There are a few subtle variations, which mostly end up producing the same kind of incredible detail. The exponentiation term can be defined by:Īnd the addition term in z -> z^n + c is similar to standard complex addition, and is simply defined by: but where 'z' and 'c' are hypercomplex ('triplex') numbers, representing Cartesian x, y, and z coordinates. The rest of the algorithm is similar to the 2D Mandelbrot!Looking at this, my brain just said to me, "Now hold on a minute Jody, do you REALLY need to do this?" No, indeed, I did not. But I searched a bit more and found this page of his, that explains things as best can be done in laymen's terms. I don't understand the maths, really, but I know more or less how it works, and that's good enough for me. So after my disillusionment, I thought "Oh well" and jumped straight into Mandelbulb 3D. Iteration I define a "pseudo power"((n)), because here, in 3D, it is not possible what was done in 2D, by the relationship X = r sin(theta)cos(phi) y = r sin(theta)sin(phi) z = r cos(theta) It happens that there are no complex numbers in a 3D equivalent, then White and Nylander performed a trick, mathematically incorrect, but graphically attractive on the 3D space coordinates to the point P, given by (x, y, z) with its spherical polar form is Thus operations, among others, exponentiation is simply: The polar form in the plane x = r cos(theta) y = r sin(theta), allows to the resort complex exponential (elementary mathematics) This definition makes it possible consistent with the usual complex operations (addition, multiplication, division, power, etc.) The first thing I decided to do was try to locate and render an image I'd seen on Daniel White's site, the "Cave of Lost Secrets" :Ĭomplex numbers are defined in the 2D plane, the expression x + iy, where i = sqrt (-1) There are quite a few more 3D Fractal programs which I'll list soon on a "Downloads" page but for our purposes, I'll use Mandelbulb 3D, written by some brilliant guy I only know, through, by the name Jesse.
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