3D Fractals

The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.

As we know, the canonical 3-dimensional Mandelbrot set doesn’t exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does (0).

The Mandelbulb is then defined as the set of those {\mathbf c} in 3 for which the orbit of \langle 0, 0, 0\rangle under the iteration {\mathbf v} \mapsto {\mathbf v}^n+{\mathbf c} is bounded. For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of “lobes” depending on n. Usually you can use n = 8 or as an example animate this POWER value.

The Sierpinski triangle is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. It is named after the Polish mathematician Wacław Sierpiński but appeared as a decorative pattern many centuries prior to the work of Sierpiński.

There are many different ways of constructing the Sierpinski triangle, this is a 3D way using raymarching.

And the last on is the Menger sponge (also known as the Menger universal curve) is a fractal curve. It is a three-dimensional generalization of the Cantor set and Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.

The Menger sponge simultaneously exhibits an infinite surface area and zero volume.

The construction of a Menger sponge can be described as follows:

  • Begin with a cube (first image).
  • Divide every face of the cube into 9 squares, like a Rubik’s Cube. This will sub-divide the cube into 27 smaller cubes.
  • Remove the smaller cube in the middle of each face, and remove the smaller cube in the very center of the larger cube, leaving 20 smaller cubes.
  • Repeat steps 2 and 3 for each of the remaining smaller cubes.

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