I've been looking at how many vertices, lines, areas, etc. that n-cubes possess. One interesting thing to fall out of these calculations is that they are continuous over the rational numbers. The number of vertices a cube in n dimensions possesses is defined as $2^d$, where d is the dimension number. My question is, outside of abstract math, what properties would non-integer dimensions possess?
For instance, what would a 1.5 dimensional circle look like? Would there be regular polytopes?