So, here is the question. Assume we have two circles. They lie in the two-dimensional space. The cartesian product of the sets representing two circles consists of 4-tuples, and thus, one may expect a geometric representation of that set in the fourth-dimensional space. Indeed, one may consider the cartesian product of a line (one-dimensional space) and a circle (two-dimensional space) as a cylindrical surface (one plus two, thus three-dimensional space). However, for two circles, one may come up with the torus representation, which seems intuitively reasonable. So, what makes two circles special in this case, and is there a general rule which allows us to determine whether there is a representation in a space of a lower dimension than the sum dimension?
P.S. I'm not sure about the tags. Sorry if they're misleading.