{\displaystyle (x,y,z)=((R+P\sin v)\cos u,(R+P\sin v)\sin u,P\cos v).} If R and P in the above flat torus parametrization form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and 0 < η < π/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 η = π/4 above, the torus will partition the 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary.
A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:

  
    
      
        (
        x
        ,
        y
        ,
        z
        ,
        w
        )
        =
        (
        R
        cos
        ⁡
        u
        ,
        R
        sin
        ⁡
        u
        ,
        P
        cos
        ⁡
        v
        ,
        P
        sin
        ⁡
        v
        )
      
    
    {\displaystyle (x,y,z,w)=(R\cos u,R\sin u,P\cos v,P\sin v)}
  where R and P are positive constants determining the aspect ratio.