One example is the torus T defined by

  
    
      
        T
        =
        
          {
          
            (
            x
            ,
            y
            ,
            z
            ,
            w
            )
            ∈
            
              
                S
              
              
                3
              
            
            ∣
            
              x
              
                2
              
            
            +
            
              y
              
                2
              
            
            =
            
              
                1
                2
              
            
            ,
             
            
              z
              
                2
              
            
            +
            
              w
              
                2
              
            
            =
            
              
                1
                2
              
            
          
          }
        
        .
{\displaystyle T=\left\{(x,y,z,w)\in \mathbf {S} ^{3}\mid x^{2}+y^{2}={\frac {1}{2}},\ z^{2}+w^{2}={\frac {1}{2}}\right\}.} 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 R4, 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 R3 was found. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals. It is a flat torus in the sense that as metric spaces, it is isometric to a flat square torus.