Third, the curvature index k determines the sign of the curvature of constant-time spatial surfaces averaged over sufficiently large length scales (greater than about a billion light-years). If k = 1, the curvature is positive and the universe has a finite volume. A universe with positive curvature is often visualized as a three-dimensional sphere embedded in a four-dimensional space. Conversely, if k is zero or negative, the universe has an infinite volume. It may seem counter-intuitive that an infinite and yet infinitely dense universe could be created in a single instant when R = 0, but exactly that is predicted mathematically when k is nonpositive and the cosmological principle is satisfied. By analogy, an infinite plane has zero curvature but infinite area, whereas an infinite cylinder is finite in one direction and a torus is finite in both.