dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be analytically embedded (smooth of class ) into Euclidean 3-space. Mapping it into 3-space requires one to stretch it, in which case it looks like a regular torus. For example, in the following map: If R and P in the above flat torus parametrization form a unit vector then u, v, and 0 < η < /2 parameterize the unit 3-sphere as Hopf coordinates. In particular, for certain very specific choices of a square flat torus in the 3-sphere S3, where above, the torus will partition the 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary. One example is the torus T defined by Other tori in S3 having this partitioning property include the square tori of the form Q⋅T, where Q is a rotation of 4-dimensional space , or in other words Q is a member of the Lie group SO(4). It is known that there exists no C2 (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of the torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem, which was proven in the 1950s, an isometric C1 embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit C1 (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space was found. It is a flat torus in the sense that as metric spaces, it is isometric to a flat square torus. It is similar in