One of the main points about what I was writing was sort of to get across how unintuitive it is. The set I wrote at the end sort of generalises to higher dimensions so if you let be the first n prime numers just to avoid factors and perfect squares and consider the set

then you have lots of functions going between -1 and 1 that are always out of phase as proved above with the irrational number bit, so they will produce every combination of n numbers between -1 and 1 and hence cover any n dimensional square/cube/ whatever you call it in higher dimensions. This is a mapping from the real numbers to that square. If you can't read the maths then try downloading the image