What is Torus: Definition and 157 Discussions

In geometry, a torus (plural tori) 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 double-covered 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.
In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, 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 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. A coffee cup and a doughnut are both topological tori.
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).

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  1. F

    Top/mani torus T^2=R^2/Z^2

    Homework Statement Consider the function f= sin(4pix)cos(6pix) on torus T^2=R^2/Z^2 a) prove this is a morse function and calculate min, max, saddle. b) describe the evolution of sublevel sets f^-1(-inf, c) as c goes from min to max Homework Equations grad(f)= <partial x, partial y> show...
  2. hideelo

    Is this a valid parametrization of the torus?

    I am reading "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo and on page 156 he gives the following parameterization of the torus x(u,v) = ((a + r cos u )cos v, (a + r cos u)sin v, r sin u) 0 < u < 2*pi, 0 < v < 2*pi Doesnt this leave out some of the torus,? I know that...
  3. O

    Surface area of a region of a torus

    What is the simplest way to calculate the surface area of a region of a torus? Please see this diagram: https://www.dropbox.com/s/73eics7x43bgiwm/surface-area-torus.png?dl=0 This is a cross section through a torus, the dashed line is the central axis. I am interested in the external surface...
  4. Symmetry777

    Concerning Quantum Phenomenon the Center of the Torus

    Concerning Quantum Phenomenon: I understand the effects of liquid nitrogen on the magnetic fields. However If you ran a tube of liquid nitrogen in the center of a torus magnet or toroidal inductor - would the magnetic field increase, toroidal magnetic vector potential increase. (Zero-point...
  5. Philip

    ± sqrt(b^2 - a^2) = ± b ± ai ?

    I came across this strange relationship when deriving the degree-4 equation for a torus. First thing that comes to mind is the 'Freshman's Dream'. Apparently, it was pure coincidence that they are equal. But, I don't believe in coincidences when it comes to a math expression. There is something...
  6. tom.stoer

    Is the twin paradox affected by the topology of a torus?

    I know that this has been discussed several times, but I was not able to find a fully convincing conclusion. Suppose there are two twins traveling with relative velocity v on a torus. 1st question: is it possible to find one unique inertial frame which can serve as a global rest frame? my...
  7. Spinnor

    Atlas of torus and sphere. Atlas of Calabi–Yau manifold.

    Is it true that the atlas for a torus can consist of a single map while the atlas for a sphere requires at least two maps? Can we ever get by with a single map for some Calabi–Yau manifolds assuming that question makes sense? If not is there some maximum number required? Thanks for any help!
  8. Salvador

    Find E Field of a Torus: Inside and Outside Surfaces

    Hello , please help me out , I can find the E field of a sphere on google and read that there is no field on the inside of the sphere , but what is the e field of a torus ? Not on the inside but on the outside surfaces also in the inner loop or the middle ?
  9. Math Amateur

    Homology Groups of the 2D Torus

    I am reading James Munkres' book, Elements of Algebraic Topology. Theorem 6.2 on page 35 concerns the homology groups of the 2-dimensional torus. Munkres shows that H_1 (T) \simeq \mathbb{Z} \oplus \mathbb{Z} and H_2 (T) \simeq \mathbb{Z} . After some work I now (just!) follow the...
  10. Math Amateur

    Torus T^2 homeomorphic to S^1 x S^1

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  11. R

    Cylindrical shells to find volume of a torus

    Homework Statement Use cylindrical shells to find the volume of a torus with radii r and R. Homework Equations V= ∫[a,b] 2πxf(x)dx y= sqrt(r2 - (x-R)2) The Attempt at a Solution V= ∫ [R, R+r] 2πx sqrt(r2 - x2 - 2xR + R2) dx I feel like this isn't going in the right direction...
  12. A

    MHB Chromatic number of a graph embedded on torus

    if the graph G can be embedded on a torus can we say: if χ(G)≥r⇒Kr≺G. Kr is a minor of G?(Kr is the complete graph with r vertices).
  13. MarkFL

    MHB Kuba's question at Yahoo Answers regarding finding the surface area of a torus

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  14. MarkFL

    MHB Roisin's question at Yahoo Answers regarding the volume of a torus

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  15. F

    Computing Natural Frequency of a Torus System: 3D vs. 2D Added Mass

    Hi! I am trying to compute the natural frequency of a system and I also need to add the added mass term to my equation. The object I have is a torus and is basically in heaving motion. I found a good paper, which gives a formula for 3d added mass and 2d added mass. I would figure that I...
  16. T

    Integral to find the volume of a torus

    Homework Statement Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the y-axis. Use geometry to evaluate the integral. Homework Equations Formula for the semi-circle: y=\sqrt{4-x^2} Solving for x give x=\pm\sqrt{4-y^2} The Attempt...
  17. L

    Parameters of helix approximated by parts of torus

    Hello, I am approximating a helix by parts of torus, to build an optical fiber wrapped around a cylinder simulation. Due to the software limitations, there is no easier way. So I take a part of the torus, rotate it so the one end points slightly up, connect similar part of the torus to the...
  18. I

    Metric on torus induced by identification of points on plane

    Hi all, Perhaps I'm asking the wrong question but I am wondering about the relationship between different definitions of, for the sake of argument, the torus. We can define it parametrically (or as a single constraint) and from there work out the induced metric as with any surface. But...
  19. N

    Mapping 4D torus onto 4-sphere non-trivially

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  20. B

    Is Air Dragged into Water When Using a Spoon with a Hole?

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  21. R

    Space as 3-Torus: Exploring Lorentz Invariance

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  22. D

    CW complex for infinite holed torus? (Surface of infinite genus)

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  23. D

    Does mass of top load torus on Tesla coil matter?

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  24. M

    Charts of a torus (and other manifolds)

    Ok, so this relates to my homework, but I really can't find an answer anywhere, so this is more of a general question. First off, what does a "chart" of a manifold look like? Is it a set, a function, a drawing, a table, what?! I have found so many things about charts, but nothing shows what...
  25. M

    How Do You Define and Use Charts on a Torus?

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  26. M

    Parameterization of a torus problem

    Homework Statement Consider the parametrization of the torus given by: x = x(s, t) = (3 + cos(s)) cos(t) y = y(s, t) = (3 + cos(s)) sin(t) z = z(s, t) = sin(s), for 0 ≤ s, t ≤ 2π. (a). What is the radius of the circle that runs though the center of the tube, and what is the radius of the tube...
  27. P

    Surface tension of a torus raindrop

    Homework Statement When calculating the difference in pressure inside a spherical raindrop, the force exerted by the surface tension is calculated to be 2pi*R*gama, where R is the radius of the drop and gamma is dE/dS (dyne/cm). When the shape of the raindrop is said to be that of a torus...
  28. W

    Mapping Torus of a Manifold is a Manifold.

    Hi, All: I'm trying to show that the Mapping torus of a manifold X is a manifold, and I'm trying to see what happens when X has a non-empty boundary B. Remember that the mapping torus M(h) of a space X by the map h is constructed like this: We start with a homeomorphism h:X-->X (we...
  29. skate_nerd

    MHB Volume of a torus, in terms of big radius R and little radius r, washer method

    This problem is really making my head spin. I'm in university calculus II and my teacher loves giving us extra homework that is supposed to really challenge us. This is problem 1 of 5. "Let 0<r<R and x2+(y-R)2=r2 be the circle centered at (0,R) of radius r. Revolving the disk enclosed by that...
  30. B

    Magnetization in a rod bent into a torus

    Homework Statement I can't figure out why there is no bound current in the problem 6 (very subtle hint boldfaced) is the pdf below: http://astronomy.mnstate.edu/cabanela/classes/phys370/homework/ps10.pdf Can anybody give me a hint as to why there should be no bound current The Attempt at a...
  31. D

    Parametric Equation of Torus: Deriving Solutions

    Homework Statement I need to derive the prarametric equation of a certain torus. defined by a unit circle on xz plane with center (a,0) and revolving about z-axis. Homework Equations * I don't know if this is relevant but here is something from wikipedia. Surfaces of revolution give another...
  32. N

    Moment of Inertia of a Torus

    Homework Statement To calculate the moment of inertia of a solid torus through the z axis(the torus is on the xy plane), using the parallel and perpendicular axis theorem. Homework Equations The Attempt at a Solution Well, first I divided the torus into tiny little disks and...
  33. L

    Is there a way to find a homeomorphism of the torus using a 2x2 integer matrix?

    For each 2x2 integer matrix with determinant +/-1 is there a homeomorphism of the 2 torus so that the the induced map on the first homology group is multiplication by this matrix?
  34. S

    Calculating Surface Area of a Torus

    So the first question is to find the surface area of a torus generated by rotating the circle (or shall I say semi circle) y=√r-x2 around y=r if the idea is to find the surface are of the half torus and then multiply by 2, wouldn't it be the same for the circle (or shall I say semi circle...
  35. F

    T-Duality, Non-Geometricity & Torus Fibrations

    Hi! I'm studying questions concerning T-duality and non-geometricity in string theory. In particular I'm focusing on the relation between T-duality and gauging of a sigma model. It's known that the free and transitive action of a set of globally defined (and commuting) Killing vectors...
  36. U

    What is the concept of a torus's center of mass and how can it be calculated?

    Homework Statement The question is attached in the picture. I did part (a) without much problems.. But I have no clue what part (b) is about at all! Even the solutions don't make much sense to me. The Attempt at a Solution I tried to work out how the diagram would look like...
  37. S

    Volume enclosed by a torus

    A torus is generated by rotating the circle x2[/SUP+(y-R)2=r2 Find the volume enclosed by the torus. Well, I don't know what to do! I thought that rewritting it as \sqrt{r2-x[SUP]2}+R would help, but I am not sure. Thanks. PD: Is this solid revolutions? because I forgot how...
  38. MathematicalPhysicist

    Morse homology of a tilted torus.

    How do I compute Morse homology of a tilted torus? Thanks.
  39. J

    Irrational Winding of the Torus

    I am trying to prove the following result: Fix a,b \in \mathbb{R} with a \neq 0. Let L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\} and let \pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2 be the canonical projection map. If \frac{b}{a} \notin \mathbb{Q}, then \pi(L) (with the subspace topology) is not a...
  40. E

    Computing the Modular Group of the Torus

    How does one compute the modular group of the torus? I see how Dehn twists generate the modular group, and I see how Dehn twists are really automorphisms of isotopy classes. Based on this, it seems that the modular group should be Aut(pi1(T^2))=Aut(Z^2)=GL(2,Z). But I've read that the modular...
  41. J

    Torus and sphere with positive or negative curvature

    Hi I know that the sphere has positive curvature everywhere, and the torus has positive an negative curvature. Is there a space homeomorphic to the sphere such that this space has negative curvature everywhere? And, is there a space homeomorphic to the torus such that the curvature is always...
  42. B

    Volumes question, volume of a torus

    Problem solved I'm not sure how easy this will be to understand without a diagram, but I don't know how to upload one :( Homework Statement Let a and b be constants, with a > b > 0.A torus is formed by rotating the circle (x - a)^2 + y^2 = b^2 about the y-axis. The cross-section at y =...
  43. A

    Torus: Mobius Strip Incisions

    whats
  44. F

    Isometric Immersion of the Torus

    Hello! I'm new here. I've been seeing this forum for a long time, but i never registered. I'd like to begin to contribute to this forum, i'll try it (even if my english doesn't help me). So now, i ask you for help: I've to find an isometric immersion of the flat torus on \mathbb{R}^4. I know...
  45. X

    Field in a superconducting torus

    A question:- Is it possible to create a magnetic field inside a perfectly superconducting torus if you did not have any to start with? I am talking about a field that goes all the way around the torus, not one that goes partway and doubles back. I am having trouble visualizing how the field...
  46. MathematicalPhysicist

    Chaotic quantum billiards on a torus.

    Hi, I want to do a similar statistics analysis as in the next paper: http://www.phy.bris.ac.uk/people/Berry_mv/the_papers/Berry340.pdf But the boundary conditions are on a two dimensioanl torus, so a solution will be of the form u(R)=\sum_{j=1}^{\infty} \sum_{m,n=0}^{\infty} (A_{mn}...
  47. T

    Electrostatics - Conducting torus and a point charge

    Homework Statement We have a metal conducting torus and a point charge that is located on the torus' axis (location on the axis is arbitrary). Calculate the (influenced) charge distribution on the torus and the electrostatic force on the point charge. Homework Equations Equation for...
  48. B

    Mapping Class Group of the Torus.

    Hi, All: I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to SL(2,Z). The method: every homeomorphism h: <\tex> T^2 -->T^2<tex> gives rise to, aka, induces an isomorphism g: <\tex> \mathbb pi_1(T^2)-->\mathbb...
  49. B

    Complement of a 'Hood of the Solid Torus in S^4

    Hi, All: This is a followup to the post : https://www.physicsforums.com/showthread.php?t=491211 Here Lavinia gave a couple of nice arguments showing that the complement of the solid torus the 3-sphere S^3 is a solid torus; one of which was using the Hopf fibration, taking a disk D^2...
  50. B

    Basic Confusion: Genus, Homology (Exm of Torus)

    Hi, All: I'm a bit confused about this: the 1st homology of the torus T(over Z) is Z(+)Z, so that elements of the form (a,b) =/ (0,0) are non-trivial , meaning these are cycles (closed curves) that do not bound subsurfaces of the torus, and every cycle that does bound is...
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