Formalization of Sierpinski's carpet and similar fractals in 2d and higher d's

[Hamzalippisch]

Hi everyone,

I have developed a set of equations to help easily implement Sierpinski's carpet, i then made variations on that equation and got other interesting fractals and even Sierpisnki's triangles. So from an IFS it is now defined by a function.

What's more is that later i generalized this equation to 3D and then to any arbitraly higher d-dimension, and the code for it remains very simple. I primarily coded it in matlab but it will work in just about any language. And finally, i have devised a way to calculate the Hausdorff dimension for any fractal of this family.

For the theory and some examples that were produced, check out the paper i made about it : https://drive.google.com/open?id=0B2ITB-CPhYwJX1VzN0Z4S2tsc0E

[DarkBeam]

It requires a permission to open it!

[Hamzalippisch]

Im sorry i thought the sharing was activated  :- it is now

[SamTiba]

I really like this one!
Nice formulas, easy implementation and stunning results
Good job and thanks for sharing your paper with us.

[mclarekin]

  This is very interesting. Would  someone write some pseudo code for 3D implementation frag??

[Hamzalippisch]

I can post some matlab code for 3D and higher, the script itself only outputs an n-dimensional matrix that you'd have to visualize after making some sections if n>3 until you have a 3D matrix then you'll need to plot it as voxels. I could've written my own voxel plotter but i downloaded a free one out of laziness  (PATCH_3DArray)

A Pseudocode for 3D would be like so :
Define precision or grid density g
Define the number of iterations iter
Define k and p (k=3 and p=3 for Menger's sponge for example)

Make 3 arrays xgrid,ygrid,zgrid from a meshgrid (0 to 1 by a step of 1/g) (3D grid coordinates)
Make a 3D matrix of ones of dimensions gxgxg, lets call it Box.

For n=1:iter
    For i=1:number of elements in Box
                Box(i)=Box(i)*sign( -sign((cos(2*pi*(k)^(n-1)*xgrid(i)))+cos(pi/(p))) - sign((cos(2*pi*(k)^(n-1)*ygrid(i)))+cos(pi/(p))) -
sign((cos(2*pi*(k)^(n-1)*zgrid(i)))+cos(pi/(p))) +3)
    End
End

Plot non-zero elements of Box as voxels