Torus
In geometry, a torus (pl.: 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
Real-world objects that approximate a torus of revolution include
A torus should not be confused with a .
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: , 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 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.[1] 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).
Etymology
Torus is a Latin word for "a round, swelling, elevation, protuberance".
Geometry
A torus can be parametrized as:[2]
using angular coordinates representing rotation around the tube and rotation around the torus' axis of revolution, respectively, where the major radius is the distance from the center of the tube to the center of the torus and the minor radius is the radius of the tube.[3]
The ratio is called the aspect ratio of the torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2.
An
Algebraically eliminating the square root gives a quartic equation,
The three classes of standard tori correspond to the three possible aspect ratios between R and r:
- When R > r, the surface will be the familiar ring torus or anchor ring.
- R = r corresponds to the horn torus, which in effect is a torus with no "hole".
- R < r describes the self-intersecting spindle torus; its inner shell is a apple
- When R = 0, the torus degenerates to the sphere.
When R ≥ r, the interior
These formulas are the same as for a cylinder of length 2πR and radius r, 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 p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q/2 and r = p − q/2), yields
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, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point.
As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, 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".[5]
In modern use,
Topology
This section includes a improve this section by introducing more precise citations. (November 2015) ) |
The surface described above, given the
The torus can also be described as a
or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA−1B−1.
The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself:
Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "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.
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.
The first
Two-sheeted cover
The 2-torus double-covers the 2-sphere, with four
n-dimensional torus
The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. (This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n.[6]) Recalling that the torus is the product space of two circles, the n-dimensional torus is the product of n circles. That is:
The standard 1-torus is just the circle: . The torus discussed above is the standard 2-torus, . And similar to the 2-torus, the n-torus, can be described as a quotient of under integral shifts in any coordinate. That is, the n-torus is modulo the
An n-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
Automorphisms of T are easily constructed from automorphisms of the lattice , which are classified by
The
Configuration space
As the n-torus is the n-fold product of the circle, the n-torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, . The configuration space of unordered, not necessarily distinct points is accordingly the orbifold , which is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates).
For n = 2, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 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.[7][8]
Flat torus
A flat torus is a torus with the metric inherited from its representation as the quotient, /L, where L is a discrete subgroup of isomorphic to . This gives the quotient the structure of a Riemannian manifold. Perhaps the simplest example of this is when L = : , which can also be described as the
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 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:
where R and P are positive constants determining the aspect ratio. It is
If R and P in the above flat torus parametrization form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and 0 < η < π/2 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 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 , 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
In April 2012, an explicit C1 (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space was found.[9][10][11][12] It is a flat torus in the sense that as metric spaces, 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. 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 obtain the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths".[13] (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 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 π and the other has 2π/3.
M may be turned into a compact space M* 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 angle around them. (Such points are termed "cusps".) This additional point will have zero 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 π/2, π/3, and 0. As a result the area of each triangle can be calculated as π - (π/2 + π/3 + 0) = π/6, so it follows that the compactified moduli space M* has area equal to π/3.
The other two cusps occur at the points corresponding in M* to a) the square torus (π) and b) the hexagonal torus (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.
Genus g surface
In the theory of surfaces there is a more general family of objects, the "genus" g surfaces. A genus g surface is the connected sum of g 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 g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles attached.
As examples, a genus zero surface (without boundary) is the
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.
genus two
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genus three
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Toroidal polyhedra
Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0. For any number of holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes.
The term "toroidal polyhedron" is also used for higher-genus polyhedra and for immersions of toroidal polyhedra.
This section needs expansion. You can help by adding to it. (April 2010) |
Automorphisms
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 of invertible integer matrices, which can be realized as linear maps on the universal covering space that preserve the standard lattice (this corresponds to integer coefficients) and thus descend to the quotient.
At the level of
Since the torus is an Eilenberg–MacLane space K(G, 1), 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
The mapping class group of higher genus surfaces is much more complicated, and an area of active research.
Coloring a torus
The torus's
de Bruijn torus
In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n 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 n is 1 (one dimension).
Cutting a torus
A solid torus of revolution can be cut by n (> 0) planes into at most
parts.[14] (This assumes the pieces may not be rearranged but must remain in place for all cuts.)
The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows:
See also
- 3-torus
- Algebraic torus
- Angenent torus
- Annulus (geometry)
- Clifford torus
- Complex torus
- Dupin cyclide
- Elliptic curve
- Irrational winding of a torus
- Joint European Torus
- Klein bottle
- Loewner's torus inequality
- Maximal torus
- Period lattice
- Real projective plane
- Sphere
- Spiric section
- Surface (topology)
- Toric lens
- Toric section
- Toric variety
- Toroid
- Toroidal and poloidal
- Torus-based cryptography
- Torus knot
- Umbilic torus
- Villarceau circles
Notes
- Nociones de Geometría Analítica y Álgebra Lineal, ISBN 978-970-10-6596-9, 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. ISBN 0-521-79540-0.
- V. V. Nikulin, I. R. Shafarevich. Geometries and Groups. Springer, 1987. ISBN 978-3-540-15281-1.
- "Tore (notion géométrique)" at Encyclopédie des Formes Mathématiques Remarquables
References
- MR 3026641.
- ^ "Equations for the Standard Torus". Geom.uiuc.edu. 6 July 1995. Archived from the original on 29 April 2012. Retrieved 21 July 2012.
- ^ "Torus". Spatial Corp. Archived from the original on 13 December 2014. Retrieved 16 November 2014.
- ^ Weisstein, Eric W. "Torus". MathWorld.
- ^ "poloidal". Oxford English Dictionary Online. Oxford University Press. Retrieved 10 August 2007.
- ^ Weisstein, Eric W. "Torus". mathworld.wolfram.com. Retrieved 27 July 2021.
- (PDF) from the original on 25 July 2011.
- ^ Phillips, Tony (October 2006). "Take on Math in the Media". American Mathematical Society. Archived from the original on 5 October 2008.
- from the original on 25 June 2012. Retrieved 21 July 2012.
- ^ Enrico de Lazaro (18 April 2012). "Mathematicians Produce First-Ever Image of Flat Torus in 3D | Mathematics". Sci-News.com. Archived from the original on 1 June 2012. Retrieved 21 July 2012.
- ^ "Mathematics: first-ever image of a flat torus in 3D – CNRS Web site – CNRS". Archived from the original on 5 July 2012. Retrieved 21 July 2012.
- ^ "Flat tori finally visualized!". Math.univ-lyon1.fr. 18 April 2012. Archived from the original on 18 June 2012. Retrieved 21 July 2012.
- ^ Hoang, Lê Nguyên (2016). "The Tortuous Geometry of the Flat Torus". Science4All. Retrieved 1 November 2022.
- ^ Weisstein, Eric W. "Torus Cutting". MathWorld.
External links
- 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 Ghostarchive and the Wayback Machine: Séquin, Carlo H (27 January 2014). "Topology of a Twisted Torus – Numberphile" (video). Brady Haran.
- Anders Sandberg (4 February 2014). "Torus Earth". Retrieved 24 July 2019.