 In particular, for certain very specific choices of a square flat torus in the 3-sphere "S"3, 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 "S"3 having this partitioning property include the square tori of the form "Q"⋅T, where "Q" is a rotation of 4-dimensional space R4, or in other words "Q" is a member of the Lie group SO(4).It is known that there exists no "C"2 (twice continuously differentiable) embedding of a flat torus into 3-space.