page_title: Torus
section_title: Flat torus
hierarchical_section_title: Torus / Flat torus
caption_attribution_description: English: Torohedron {4,4}_1,0, 1 vertex, 2 monogonal edges, and 1 square face
context_page_description: In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
  If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a sphere. If the revolved curve is not a circle, the surface is a related shape, a toroid.
  Real-world objects that approximate a torus of revolution include swim rings and inner tubes. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses.
  A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.
context_section_description: The flat torus is a torus with the metric inherited from its representation as the quotient, R²/L, where L is a discrete subgroup of R² isomorphic to Z². This gives the quotient the structure of a Riemannian manifold. Perhaps the simplest example of this is when L = Z²: R²/Z², which can also be described as the Cartesian plane under the identifications (x, y) ~ (x + 1, y) ~ (x, y + 1). This particular flat torus (and any uniformly scaled version of it) is known as the "square" flat torus.
  This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is "flat" in the same sense that the surface of a cylinder is "flat". In 3 dimensions one can bend a flat sheet of paper into a cylinder without stretching the paper, but you cannot then bend this cylinder into a torus without stretching the paper (unless you give up some regularity and differentiability conditions, 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 constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be analytically embedded (smooth of class Cᵏ, 2 ≤ k ≤ ∞) into Euclidean 3-space. Mapping it into 3-space requires you to stretch it, in which case it looks like a regular torus, for example, the following map
  If R and P in the above flat torus form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and η can be used to parameterize the unit 3-sphere in a parameterization associated with the Hopf map. 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. 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 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, 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. (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations, or depicted by computer graphics.
caption: The simplest tiling of a flat torus is {4,4}1,0, constructed on the surface of a duocylinder with 1 vertex, 2 orthogonal edges, and one square face. It is seen here stereographically projected into 3-space as a torus.
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