In fact, S3 is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S3 as a fiber bundle over S2 (the Hopf bundle). The surface described above, given the relative topology from R3, is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R3 from the north pole of S3.
In fact, the conformal type of the torus is determined by the cross-ratio of the four points. n-dimensional torus
The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. (This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n.) Recalling that the torus is the product space of two circles, the n-dimensional torus is the product of n circles.