Editing Torus (section)

== Flat torus ==

[[File:Torus from rectangle.gif|thumb|In three dimensions, one can bend a rectangle into a torus, but doing this necessarily affects the distances measured along the surface, as seen by the distortion of the checkered pattern.]]
[[File:Duocylinder ridge animated.gif|thumb|Seen in [[stereographic projection]], a 4D ''flat torus'' can be projected into 3-dimensions and rotated on a fixed axis.]]
[[File:Toroidal monohedron.png|thumb|The simplest tiling of a flat torus is [[regular map (graph theory)#Toroidal polyhedra|{4,4}<sub>1,0</sub>]], 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.]]
A flat torus is a torus with the metric inherited from its representation as the [[quotient space (topology)|quotient]], {{math|1='''R'''<sup>2</sup> / '''L'''}}, where {{math|'''L'''}} is a discrete subgroup of {{math|1='''R'''<sup>2</sup>}} isomorphic to {{math|1='''Z'''<sup>2</sup>}}. This gives the quotient the structure of a [[Riemannian manifold]], as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when {{math|1='''L''' = '''Z'''<sup>2</sup>}}: {{math|'''R'''<sup>2</sup> / '''Z'''<sup>2</sup>}}, which can also be described as the [[Cartesian plane]] under the identifications {{math|(''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. It 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 this cylinder cannot be bent into a torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below).

A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows:
: <math>(x,y,z,w) = (R\cos u, R\sin u, P\cos v, P\sin v)</math>
where ''R'' and ''P'' are positive constants determining the aspect ratio. It is [[diffeomorphism|diffeomorphic]] to a regular torus but not [[isometry|isometric]]. It can not be [[analytic function|analytically]] embedded ([[smooth function|smooth]] of class {{math|''C<sup>k</sup>'', 2 ≤ ''k'' ≤ ∞}}) into Euclidean 3-space. [[Map (mathematics)|Mapping]] it into ''3''-space requires one to stretch it, in which case it looks like a regular torus. For example, in the following map:
: <math>(x,y,z) = ((R+P\sin v)\cos u, (R+P\sin v)\sin u, P\cos v).</math>

If {{math|''R''}} and {{math|''P''}} in the above flat torus parametrization form a unit vector {{math|1=(''R'', ''P'') = (cos(''η''), sin(''η''))}} then ''u'', ''v'', and {{math|0 < ''η'' < π/2}} parameterize the unit 3-sphere as [[3-sphere#Hopf coordinates|Hopf coordinates]]. In particular, for certain very specific choices of a square flat torus in the [[3-sphere]] ''S''<sup>3</sup>, where {{math|1=''η'' = π/4}} above, the torus will partition the 3-sphere into two [[congruence (geometry)|congruent]] solid tori subsets with the aforesaid flat torus surface as their common [[boundary (topology)|boundary]]. One example is the torus {{math|''T''}} defined by
: <math>T = \left\{ (x,y,z,w) \in S^3 \mid x^2+y^2 = \frac 1 2, \ z^2+w^2 = \frac 1 2 \right\}.</math>

Other tori in {{math|''S''<sup>3</sup>}} having this partitioning property include the square tori of the form {{math|''Q'' ⋅ ''T''}}, where {{math|''Q''}} is a rotation of 4-dimensional space {{math|'''R'''<sup>4</sup>}}, or in other words {{math|''Q''}} is a member of the Lie group {{math|SO(4)}}.

It is known that there exists no {{math|''C''<sup>2</sup>}} (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 embedding theorem|Nash-Kuiper theorem]], which was proven in the 1950s, an isometric ''C''<sup>1</sup> embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding.

[[File:Flat torus Havea embedding.png|thumb|right|<math>C^1</math> isometric embedding of a flat torus in {{math|'''R'''<sup>3</sup>}}, with corrugations]]
In April 2012, an explicit ''C''<sup>1</sup> (continuously differentiable) isometric embedding of a flat torus into 3-dimensional Euclidean space {{math|'''R'''<sup>3</sup>}} was found.<ref>{{cite journal|last=Filippelli|first=Gianluigi|date=27 April 2012|title=Doc Madhattan: A flat torus in three dimensional space|url=http://docmadhattan.fieldofscience.com/2012/04/flat-torus-in-three-dimensional-space.html|url-status=live|journal=Proceedings of the National Academy of Sciences|volume=109|issue=19|pages=7218–7223|doi=10.1073/pnas.1118478109|pmc=3358891|pmid=22523238|archive-url=https://web.archive.org/web/20120625222341/http://docmadhattan.fieldofscience.com/2012/04/flat-torus-in-three-dimensional-space.html|archive-date=25 June 2012|access-date=21 July 2012|doi-access=free}}</ref><ref>{{cite web|url=http://www.sci-news.com/othersciences/mathematics/article00279.html|title=Mathematicians Produce First-Ever Image of Flat Torus in 3D {{pipe}} Mathematics|last=Enrico de Lazaro|date=18 April 2012|website=Sci-News.com|url-status=live|archive-url=https://web.archive.org/web/20120601021059/http://www.sci-news.com/othersciences/mathematics/article00279.html|archive-date=1 June 2012|access-date=21 July 2012}}</ref><ref>{{cite web|url=http://www2.cnrs.fr/en/2027.htm|title=Mathematics: first-ever image of a flat torus in 3D – CNRS Web site – CNRS|url-status=dead|archive-url=https://web.archive.org/web/20120705120058/http://www2.cnrs.fr/en/2027.htm|archive-date=5 July 2012|access-date=21 July 2012}}</ref><ref>{{cite web |url=http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html |title=Flat tori finally visualized! |publisher=Math.univ-lyon1.fr |date=18 April 2012 |access-date=21 July 2012 |url-status=dead |archive-url=https://web.archive.org/web/20120618084643/http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html |archive-date=18 June 2012}}</ref> It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a [[fractal]] as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined [[normal (geometry)|surface normals]], yielding a so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths".<ref>{{cite web |url=http://www.science4all.org/article/flat-torus/ |title=The Tortuous Geometry of the Flat Torus |last=Hoang |first=Lê Nguyên |date=2016 |website=Science4All |access-date=November 1, 2022}}</ref> (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.

=== Conformal classification of flat tori ===

In the study of [[Riemann surface|Riemann surfaces]], one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The [[Uniformization theorem]] guarantees that every Riemann surface is [[Conformal map|conformally equivalent]] to one that has constant [[Gaussian curvature]]. In the case of a torus, the constant curvature must be zero. Then one defines the "[[moduli space]]" of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space ''M'' may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle π and the other has total angle 2π/3. 

''M'' may be turned into a compact space ''M*'' – topologically equivalent to a sphere – by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space  is a sphere with ''three'' points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, ''M*'' may be constructed by glueing together two congruent [[geodesic triangle]]s in the [[hyperbolic plane]] along their (identical) boundaries, where each triangle has angles of {{math|π/2}}, {{math|π/3}}, and {{math|0}}. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the [[Gauss–Bonnet theorem]] shows that the area of each triangle can be calculated as {{math|1=π − (π/2 + π/3 + 0) = π/6}}, so it follows that the compactified moduli space ''M*'' has area equal to {{math|π/3}}.

The other two cusps occur at the points corresponding in ''M*'' to (a) the square torus (total angle {{math|π}}) and (b) the hexagonal torus (total angle {{math|2π/3}}). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation.