Is it possible to do something similar on an n-cube for n>3 ?
There’s no mention of that in the wiki article so maybe I’m missing something obvious (do n-dimensional fractals exist in general?).
Comment by Craig — Saturday 5 August 2006 7:58 am #
Yes – it is possible. There’s a 4D HyperGasket which also says: By following the procedure of creating a gasket in the 2nd and 3rd dimensions it is possible to do the same thing in higher dimensions, in particular the 4th dimension
and the recipe for a 4D version made from a hypercube is at Hyperspace Travel Guide. These are really just Cantor’s Set in higher dimensions so it is easy to see how it can be done there.
A picture of a stage-6 Menger Sponge can be found at Hexahedron Fractal. It is interesting to record that the Menger Sponge has dimension 2.73 (more precisely ).
Comment by Steve — Saturday 5 August 2006 10:26 am #
A ha! Thanks for the links. Yes, it’s interesting to note that what you intuitively think of as a 3-dimensional thing is actually 2.73 dimensional and so on for higher dimensions.