{\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.
The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane with itself. This produces a geometric object called the Clifford torus, a surface in 4-space. In the field of topology, a torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one. An example of a torus can be constructed by taking a rectangular strip of flexible material, for example, a rubber sheet, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Möbius strip). Geometry
A torus can be defined parametrically by:
where

θ, φ are angles which make a full circle, so their values start and end at the same point,
R is the distance from the center of the tube to the center of the torus,
r is the radius of the tube.Angle θ represents rotation around the tube, whereas φ represents rotation around the torus axis of revolution. R is known as the "major radius" and r is known as the "minor radius". The ratio R divided by r is known as the "aspect ratio". The typical doughnut confectionery has an aspect ratio of about 3 to 2.