# Ways to project arbitrary Fractals on 2D objects and 3D objects w different dimensions?

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

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.