![]() ![]() Analytically, fractals are usually nowhere differentiable. ![]() However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two the ratio of the new to the old radius to the power of three the dimension that the sphere resides in. Doubling the edge lengths of a polygon multiplies its area by four, which is two the ratio of the new to the old side length raised to the power of two the dimension of the space the polygon resides in. One way that fractals are different from finite geometric figures is the way in which they scale. Mandelbulb3D/Reference/Formulas/Formula types Fractals exhibit similar patterns at increasingly small scales called self similarityalso known as expanding symmetry or unfolding symmetry if this replication is exactly the same at every scale, as in the Menger sponge it is called affine self-similar.įractal geometry lies within the mathematical branch of measure theory. The names of the parameter files match the names of the images.The images are also included in the zip file for reference.In mathematicsa fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension.įractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set because of this, fractals are encountered ubiquitously in nature. ![]() m3p files that can be loaded and explored in Mandelbulb 3D. ![]() Mouse-over each image to see the filename.Īll of these use Amazing Box, _, Folding Int Pow, and ABoxVaryScale4d formulas with identical parameters, except for the blank which is filled in by whichever formula I was testing at the moment.ĭownload the zip file that contains this set of parameters. The images below also may give you an idea of what the different transformations can do. There are plenty of interesting bits to explore in these images, far too many for me to ever do justice to all the possibilities. There are one or two variations for each formula. The results of these experiments are yours to use and expand on. Last night, starting with this Mandelbulb 3D file, I spent a few hours experimenting with substituting different formulas in one position, holding all other parameters the same. ![]()
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