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I am trying to create a house/texture in 3D and in 2D with fractals, perhaps related. My friend said that fractals can have different dimensions such as 1.74, 1, 4.71111... and pretty much anything. Now I want to create a castle from some fractal such as Mandelbulb fractal. Some fractals have the 3D presentation such as the spherical-form of the mandelbulb $$\left<x,y,z\right>=\left<\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta)\right>$$ Now suppose we have some fractal that does not have 3D presentation.

What kind of ways can be used to project an arbitrary fractal on the surface of a let say texture, a cube or arbitrary surface?

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To project arbitrary fractal onto arbitrary surface of arbitrary dimension use arbitrary projection. – dtldarek Mar 18 '13 at 22:15
@dtldarek such projection does not exist? Any theorem to project let say some 2.123 dimension fractal onto 2D plane? I am still trying to learn this concept of non-integer dimensions... – hhh Mar 18 '13 at 22:17
up vote 1 down vote accepted

The non-integral fractional dimensions usually won't have to concern you.

Take for example the Sierpinski triangle. If you double the edge length, you get three copies of the original triangle. Compare this to a simple (i.e. non-fractal) equilateral triangle: if you double its edge length, you quadruple its area. So the triangle has dimension $$\frac{\log 4}{\log 2}=2$$ which tells you that the triangle is a normal area, more than a line (or path) segment but less than a volume. Now do the same for the Sierpinski triangle, and you get a dimension of $$\frac{\log 3}{\log 2}\approx1.5849625$$ so it is in a certain sense more than a line but less than a fully covered area. Which makes sense, because you couldn't reasonably map it onto a line, but if you choose a point randomly you can be “almost certain” that it will fall into a hole at some level of the iteration. So this is the reason why one can associate a non-integral dimension with the Sierpinski triangle, in a rather intuitive description of the Hausdorff dimension.

This does not change the fact that the Sierpinski triangle, by its definition, is embedded into the plane. So when mapping this in some geometric sense, you'd usually start with that plane, and do whatever you are used to doing to planes. I know of no fractal which does not come with a reasonable embedding into some integer-dimensioned Euclidean vector space.

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