Torus
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In geometry, a torus (Template:Plural form: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. 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 (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a toroid, as in a square toroid.
Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings.
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.
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: Template:Math, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. 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 Template:Math 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.<ref> Template:Cite book</ref> 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 such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Klein bottle).
EtymologyEdit
Torus is a Latin word denoting something round, a swelling, an elevation, a protuberance.
GeometryEdit
A torus of revolution in 3-space can be parametrized as:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> <math display="block">\begin{align} x(\theta, \varphi) &= (R + r \sin \theta) \cos{\varphi}\\ y(\theta, \varphi) &= (R + r \sin \theta) \sin{\varphi}\\ z(\theta, \varphi) &= r \cos \theta\\ \end{align}</math> using angular coordinates Template:Math, Template:Math, representing rotation around the tube and rotation around the torus's axis of revolution, respectively, where the major radius Template:Math is the distance from the center of the tube to the center of the torus and the minor radius Template:Math is the radius of the tube.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
The ratio Template:Math is called the aspect ratio of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2.
An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is <math display="block">{\textstyle \bigl(\sqrt{x^2 + y^2} - R\bigr)^2} + z^2 = r^2.</math>
Algebraically eliminating the square root gives a quartic equation, <math display="block">\left(x^2 + y^2 + z^2 + R^2 - r^2\right)^2 = 4R^2\left(x^2+y^2\right).</math>
The three classes of standard tori correspond to the three possible aspect ratios between Template:Mvar and Template:Mvar:
- When Template:Math, the surface will be the familiar ring torus or anchor ring.
- Template:Math corresponds to the horn torus, which in effect is a torus with no "hole".
- Template:Math describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple.
- When Template:Math, the torus degenerates to the sphere radius Template:Math.
- When Template:Math, the torus degenerates to the circle radius Template:Math.
When Template:Math, the interior <math display="block">{\textstyle \bigl(\sqrt{x^2 + y^2} - R\bigr)^2} + z^2 < r^2</math> of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem, giving:<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Torus%7CTorus.html}} |title = Torus |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> <math display="block">\begin{align} A &= \left( 2\pi r \right) \left(2 \pi R \right) = 4 \pi^2 R r, \\[5mu] V &= \left ( \pi r^2 \right ) \left( 2 \pi R \right) = 2 \pi^2 R r^2. \end{align}</math>
These formulae are the same as for a cylinder of length Template:Math and radius Template:Mvar, obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
Expressing the surface area and the volume by the distance Template:Mvar of an outermost point on the surface of the torus to the center, and the distance Template:Mvar of an innermost point to the center (so that Template:Math and Template:Math), yields <math display="block">\begin{align} A &= 4 \pi^2 \left(\frac{p+q}{2}\right) \left(\frac{p-q}{2}\right) = \pi^2 (p+q) (p-q), \\[5mu] V &= 2 \pi^2 \left(\frac{p+q}{2}\right) \left(\frac{p-q}{2}\right)^2 = \tfrac14 \pi^2 (p+q) (p-q)^2. \end{align}</math>
As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, Template:Mvar, the distance from the center of the coordinate system, and Template:Mvar and Template:Mvar, angles measured from the center point.
As a torus has, effectively, two center points, the centerpoints of the angles are moved; Template:Mvar measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of Template:Mvar is moved to the center of Template:Mvar, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices.
TopologyEdit
Template:No footnotes Topologically, a torus is a closed surface defined as the product of two circles: Template:Math. This can be viewed as lying in [[complex coordinate space|Template:Math]] and is a subset of the 3-sphere Template:Math of radius Template:Math. This topological torus is also often called the Clifford torus.<ref>Template:Cite journal</ref> In fact, Template:Math is filled out by a family of nested tori in this manner (with two degenerate circles), a fact that is important in the study of Template:Math as a fiber bundle over Template:Math (the Hopf bundle).
The surface described above, given the relative topology from [[real coordinate space|Template:Math]], is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into Template:Math from the north pole of Template:Math.
The torus can also be described as a quotient of the Cartesian plane under the identifications
- <math>(x,y) \sim (x+1,y) \sim (x,y+1), \,</math>
or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon Template:Math.
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The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:
- <math>\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathrm{Z} \times \mathrm{Z}.</math><ref>Padgett, Adele (2014). "Fundamental groups: motivation, computation methods, and applications", REA Program, Uchicago. https://math.uchicago.edu/~may/REU2014/REUPapers/Padgett.pdf</ref>
Intuitively speaking, this means that a closed path that circles the torus's "hole" (say, a circle that traces out a particular latitude) and then circles the torus's "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding.
The fundamental group can also be derived from taking the torus as the quotient <math>T^2\cong \mathbb{R}^2/\mathbb{Z}^2</math> (see below), so that <math>\mathbb{R}^2</math> may be taken as its universal cover, with deck transformation group <math>\mathbb{Z}^2=\pi_1(T^2)</math>.
Its higher homotopy groups are all trivial, since a universal cover projection <math>p:\widetilde{X}\rightarrow X</math> always induces isomorphisms between the groups <math>\pi_n(\widetilde{X})</math> and <math>\pi_n(X)</math> for <math>n>1</math>, and <math>\mathbb{R}^2</math> is contractible.
The torus has homology groups
<math>H_n(T^2)=\begin{cases}\mathbb{Z},& n=0,2\\
\mathbb{Z}\oplus \mathbb{Z},& n=1\\
0&\text{else.}\end{cases}</math>
Thus, the first homology group of the torus is isomorphic to its fundamental group-- which in particular can be deduced from Hurewicz theorem since <math>\pi_1(T^2)</math> is abelian.
The cohomology groups with integer coefficients are isomorphic to the homology ones-- which can be seen either by direct computation, the universal coefficient theorem or even Poincaré duality.
If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.
Two-sheeted coverEdit
The 2-torus is a twofold branched cover of the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as such a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of the torus is determined by the cross-ratio of the four points.
n-dimensional torusEdit
The torus has a generalization to higher dimensions, the Template:Em, often called the Template:Em or Template:Em for short. (This is the more typical meaning of the term "Template:Math-torus", the other referring to Template:Math holes or of genus Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>) Just as the ordinary torus is topologically the product space of two circles, the Template:Math-dimensional torus is topologically equivalent to the product of Template:Math circles. That is:
- <math>T^n = \underbrace{S^1 \times \cdots \times S^1}_n.</math>
The standard 1-torus is just the circle: Template:Math. The torus discussed above is the standard 2-torus, Template:Math. And similar to the 2-torus, the Template:Math-torus, Template:Math can be described as a quotient of Template:Math under integral shifts in any coordinate. That is, the n-torus is Template:Math modulo the action of the integer lattice Template:Math (with the action being taken as vector addition). Equivalently, the Template:Math-torus is obtained from the Template:Math-dimensional hypercube by gluing the opposite faces together.
An Template:Math-torus in this sense is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group Template:Math one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori Template:Math have a controlling role to play in theory of connected Template:Math. Toroidal groups are examples of protori, which (like tori) are compact connected abelian groups, which are not required to be manifolds.
Automorphisms of Template:Math are easily constructed from automorphisms of the lattice Template:Math, which are classified by invertible integral matrices of size Template:Math with an integral inverse; these are just the integral matrices with determinant Template:Math. Making them act on Template:Math in the usual way, one has the typical toral automorphism on the quotient.
The fundamental group of an n-torus is a free abelian group of rank Template:Math. The Template:Mathth homology group of an Template:Math-torus is a free abelian group of rank n choose Template:Math. It follows that the Euler characteristic of the Template:Math-torus is Template:Math for all Template:Math. The cohomology ring H•(<math>T^{n}</math>, Z) can be identified with the exterior algebra over the Template:Math-module Template:Math whose generators are the duals of the Template:Math nontrivial cycles.
Configuration spaceEdit
As the Template:Math-torus is the Template:Math-fold product of the circle, the Template:Math-torus is the configuration space of Template:Math ordered, not necessarily distinct points on the circle. Symbolically, Template:Math. The configuration space of unordered, not necessarily distinct points is accordingly the orbifold Template:Math, which is the quotient of the torus by the symmetric group on Template:Math letters (by permuting the coordinates).
For Template:Math, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For Template:Math this quotient may be described as a solid torus with cross-section an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.
These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads.<ref>Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Flat torusEdit
A flat torus is a torus with the metric inherited from its representation as the quotient, Template:Math, where Template:Math is a discrete subgroup of Template:Math isomorphic to Template:Math. 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 Template:Math: Template:Math, which can also be described as the Cartesian plane under the identifications Template:Math. 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 diffeomorphic to a regular torus but not isometric. It can not be analytically embedded (smooth of class Template:Math) into Euclidean 3-space. 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 Template:Math and Template:Math in the above flat torus parametrization form a unit vector Template:Math then u, v, and Template:Math 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 Template:Math 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 Template:Math 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 Template:Math having this partitioning property include the square tori of the form Template:Math, where Template:Math is a rotation of 4-dimensional space Template:Math, or in other words Template:Math is a member of the Lie group Template:Math.
It is known that there exists no Template:Math (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) isometric embedding of a flat torus into 3-dimensional Euclidean space Template:Math was found.<ref>Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 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>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 toriEdit
In the study of 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 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 triangles in the hyperbolic plane along their (identical) boundaries, where each triangle has angles of Template:Math, Template:Math, and Template:Math. (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 Template:Math, so it follows that the compactified moduli space M* has area equal to Template:Math.
The other two cusps occur at the points corresponding in M* to (a) the square torus (total angle Template:Math) and (b) the hexagonal torus (total angle Template:Math). These are the only conformal equivalence classes of flat tori that have any conformal automorphisms other than those generated by translations and negation.
Genus g surfaceEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In the theory of surfaces there is a more general family of objects, the "genus" Template:Math surfaces. A genus Template:Math surface is the connected sum of Template:Math two-tori. (And so the torus itself is the surface of genus 1.) To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary circles. (That is, merge the two boundary circles so they become just one circle.) To form the connected sum of more than two surfaces, successively take the connected sum of two of them at a time until they are all connected. In this sense, a genus Template:Math surface resembles the surface of Template:Math doughnuts stuck together side by side, or a 2-sphere with Template:Math handles attached.
As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called Template:Math-holed tori (or, rarely, Template:Math-fold tori). The terms double torus and triple torus are also occasionally used.
The classification theorem for surfaces states that every compact connected surface is topologically equivalent to either the sphere or the connect sum of some number of tori, disks, and real projective planes.
| File:Double torus illustration.pngTemplate:Brgenus two | File:Triple torus illustration.pngTemplate:Brgenus three |
Toroidal polyhedraEdit
Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic Template:Math. For any number of holes, the formula generalizes to Template:Math, where Template:Math is the number of holes.
The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra.
AutomorphismsEdit
The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group (the connected components of the homeomorphism group) is surjective onto the group <math>\operatorname{GL}(n,\mathbf{Z})</math> of invertible integer matrices, which can be realized as linear maps on the universal covering space <math>\mathbf{R}^{n}</math> that preserve the standard lattice <math>\mathbf{Z}^{n}</math> (this corresponds to integer coefficients) and thus descend to the quotient.
At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra:
- <math>\operatorname{MCG}_{\operatorname{Ho}}(T^n) = \operatorname{Aut}(\pi_1(X)) = \operatorname{Aut}(\mathbf{Z}^n) = \operatorname{GL}(n,\mathbf{Z}).</math>
Since the torus is an Eilenberg–MacLane space Template:Math, its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); all homotopy equivalences of the torus can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism.
Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of <math>\mathbf{R}^{n}</math> gives a splitting, via the linear maps, as above):
- <math>1 \to \operatorname{Homeo}_0(T^n) \to \operatorname{Homeo}(T^n) \to \operatorname{MCG}_{\operatorname{TOP}}(T^n) \to 1.</math>
The mapping class group of higher genus surfaces is much more complicated, and an area of active research.
Coloring a torusEdit
The torus's Heawood number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven. (Since the complete graph <math>\mathsf{K_7}</math> can be embedded on the torus, and <math>\chi (\mathsf{K_7}) = 7</math>, the upper bound is tight.) Equivalently, in a torus divided into regions, it is always possible to color the regions using no more than seven colors so that no neighboring regions are the same color. (Contrast with the four color theorem for the plane.)
de Bruijn torusEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet (often just 0 and 1) that contains every Template:Math-by-Template:Math matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where Template:Math is 1 (one dimension).
Cutting a torusEdit
A solid torus of revolution can be cut by n (> 0) planes into at most
- <math>\begin{pmatrix}n+2 \\ n-1\end{pmatrix} +\begin{pmatrix}n \\ n-1\end{pmatrix} = \tfrac{1}{6}(n^3 + 3n^2 + 8n)</math>
parts.<ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:TorusCutting%7CTorusCutting.html}} |title = Torus Cutting |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref> (This assumes the pieces may not be rearranged but must remain in place for all cuts.)
The first 11 numbers of parts, for Template:Math (including the case of Template:Math, not covered by the above formulas), are as follows:
See alsoEdit
Template:Portal Template:Columns-list
NotesEdit
- Nociones de Geometría Analítica y Álgebra Lineal, Template:ISBN, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish
- Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. Template:ISBN.
- V. V. Nikulin, I. R. Shafarevich. Geometries and Groups. Springer, 1987. Template:ISBN, Template:ISBN.
- "Tore (notion géométrique)" at Encyclopédie des Formes Mathématiques Remarquables
ReferencesEdit
External linksEdit
Template:Sister project Template:Commons and category
- Creation of a torus at cut-the-knot
- "4D torus" Fly-through cross-sections of a four-dimensional torus
- "Relational Perspective Map" Visualizing high dimensional data with flat torus
- Polydoes, doughnut-shaped polygons
- Archived at GhostarchiveTemplate:Cbignore and the Wayback MachineTemplate:Cbignore: {{#invoke:citation/CS1|citation
|CitationClass=web }}Template:Cbignore
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