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. D

    Homology of torus and Klein's bottle

    Homework Statement I'm trying to calculate singular homology groups of the torus and Klein's bottle using the Mayer-Vietoris sequence. The Attempt at a Solution I represent both spaces as a rectangle with identified edges. Then I take the sets: U=rectangle without the boundary...
  2. S

    Proving the parametrization of a Torus imbedded in R3 is a Quotient map

    Homework Statement Let b > a > 0. Consider the map F : [0, 1] X [0, 1] -> R3 defined by F(s, t) = ((b+a cos(2PIt)) cos(2PIs), (b+a cos(2PIt)) sin(2PIs), a sin(2PIt)). This is the parametrization of a Torus. Show F is a quotient map onto it's image. Homework Equations Proving that any subset...
  3. R

    Calculate Volume of Partial Torus (a=10, r=5)

    I have a real world problem. To design a calculator for a special machining process, I need to determine the volume of material removed when making a groove in the shape of a semicircle inside a tube on a lathe. The volume removed looks like this...
  4. B

    What is the connection between the mapping class group of a torus and Gl(2,Z)?

    Hi, everyone: I am trying to understand why the mapping class group of the torus T^2 (i.e., the group of orientation-preserving self- diffeomorphisms, up to isotopy) is (iso. to) Gl(2,Z) ( I just realized this is the name of the group of orientation-preserving...
  5. L

    Why the crystal momentum k in 2D band structure is defined on a torus?

    Hi, Could someone help to explain that why for the 2 D system, like 2 dimensional electronic gas, the crystal momentum k in 2D band structure is defined on a torus? Thanks a lot.:)
  6. L

    Calculating Force on a Torus: Lennard Jones Potential

    The problem is with calculating the force of a particle by another particle while both particles lie on a torus. By this I mean, if particle i and j are in a box (LxL) and particle i is on the left edge of the box and particle j is on the right edge of the box, the distance used in the force...
  7. L

    Why Do Argon Atoms in a Toroidal Box Show Unusually Large Positional Values?

    So I'm working on a problem where you have 100 argon atoms in a box that obey the Lennard Jones potential. We're using the Verlet method to calculate the position of each particle at the next time step. The issue I'm having has to do with our boundary conditions. We assume that atoms are on a...
  8. G

    How to prove that a given set is a torus

    Hi I'm trying to solve this exercise "Prove that if C is a circular cylinder with S_1 and S_2 as its boundary circles and S_1 and S_2 are identified by mapping them both homeomorphically onto some third circle S, giving a map f: S_1 \cup S_2 \rightarrow S then (C - S_1 \cup S_2) \cup S...
  9. M

    Surface Integrals and Average Surface Temperature of a Torus

    Homework Statement A torus is a surface obtained by rotating a circle about a straight line. (It looks like a doughnut.) If the z-axis is the axis of rotation and the circle has radius b, centre (0, a, 0) with a > b, and lies in y − z plane, the torus obtained has the parametric form r(u, v) =...
  10. K

    What's the fundamental group of a punctured torus?

    The fundamental group of a torus is Z*Z,then the fundamental group of a punctured torus is Z*Z*Z. But I've ever done a problem,it said a punctured torus can be continuously deformed into two cylinders glued to a square patch.Really? If that is right,then the fundamental group of punctured...
  11. N

    Solids of rotation(volume of a torus)

    The problem reads(from Stewart Calculus Concepts and Contexts 4th edition, Ch.6 section 2 pg. 447 #45 a)Set up an integral for the volume of a solid torus(the donut-shaped solid shown in the figure) with radii r and R b)By interpreting the integral as an area, find the volume of the torus...
  12. L

    Twin Paradox in a Flat Toroidal Universe: Time Dilation and Inertial Frames

    In four dimensions, a flat torus is an object that has zero curvature but still has closed geodesic curves. What this means is that if you try to measure geometry locally, you will find that it is perfectly Euclidean. Nevertheless, if you travel on a straight line, you'll eventually end up...
  13. J

    Understanding Torus Parameterization

    Homework Statement Consider the parametrization of torus given by: x=x(ø,ß)=(3+cos(ø))cos(ß) y=y(ø,ß)=(3+cos(ø))sin(ß) z=z(ø,ß)=sin(ø), for 0≤ø,ß≤2π What is the radius of the circle that runs through the center of the tube, and what is the radius of the tube, measured from the...
  14. L

    What is the concept of exact forms on a 2 torus and how can it be visualized?

    I am having trouble visualizing when a 2 form is exact and have a specific case that I am struggling with at the moment. Any help is welcome. Take an oriented 2 torus and divide it ,using parallel circles, into an even number of tube shaped regions. In each tube, assign a 2-form that fades...
  15. E

    Unlink a Two-Holed Torus in 3D: Impossible or Possible?

    1. Explain how a two-holed Torus with it's holes linked can be distorted into an unlinked two holed torus 2. One Linked Genus 2 Torus to be distorted into an Unlinked Two holed Torus 3. It sems impossible to unlink it in #D. So I thought maybe lifting one link into the fourth...
  16. B

    Triangulation of Torus, Algorithms for Calculating Simplicial Homology

    Hi, everyone: A couple of points, please: 1) I am reviewing last semester's Simplicial Homology. I was able to do a triangulation of the torus T2=S1xS1 , and I was able to do a triangulation of T2 , although the best I could do was use 18 triangles. (the...
  17. B

    N-dimensional torus compactification and string winding

    Hello, Im trying to learn about string theory in toroidal compactification on an undergraduate level. I am mostly using Zwiebach's "A first course in string Theory" but now I am trying to do something that doesn't seem to be covered in the book or any other literature explicitly. Perhaps...
  18. C

    How to make curved line into torus?

    Greetings, I am trying to figure out how to modify the equations for a trefoil knot to make it toroidal. A trefoil knot is: x = (2 + cos 3t)cos 2t y = (2 + cos 3t)sin 2t z = sin 3t A torus is: x(u, v) = (R + r cos v) cos u y(u, v) = (R + r cos v) sin u z(u,v) = r sin v where...
  19. L

    Define the mapping torus of a homeomorphism

    Define the mapping torus of a homeomorphism \phi:X \rightarrow X to be the identification space T(\phi)= X \times I / \{ (x,0) \sim (\phi(x),1) | x \in X \} I have to identify T(\phi) with a standard space and prove that it is homotopy equivalent to S^1 by constructing explicit maps f:S^1...
  20. A

    Fundemental group, Torus, SVK

    Hi So I've been using Seifert-Van Kampen (SVK) to calculate the fundamental group of the torus. I haven't done any formal group theory, hence my problem ... I have T^2=(S^1)x(S^1) If A= S^1, B=S^1, A intersection B is 0. And T^2 = union of A and B. Then fundamental group of (A...
  21. B

    Find an atlas and coordinates for a torus T^2 = S^1 X S^1

    Homework Statement Find an atlas and coordinates for a torus T^2 = S^1 X S Homework Equations The Attempt at a Solution I know that an atlas on a manifold M is a collection of charts whose domains cover M, but i am not sure how to start this one mathematically.
  22. P

    Hypothetical space station orbiting inside a plasma torus

    Consider, if it were the case, that the plasma torus left in Io's path as it orbits Jupiter followed the same orbital path as the moon and was not deformed by Jupiter's magnetosphere...thus a space station orbiting Jupiter could effectively orbit in Io's plasma torus. Yes, this is going...
  23. M

    Two torus identification problems

    1. Compute the fundamental group of the space obtained from 2 two-dimensional tori S1 x S1 by identifying a circle S1 x {0} in one torus with the corresponding circle S1 x {0} in the other torus. Am I right in thinking that the resulting space is S1 x S1 x S1, and hence the fundamental group...
  24. Spinnor

    1D string on a cylinder and torus.

    The physics of a 1D string with fixed end points is found here: http://www.uio.no/studier/emner/matnat/ifi/INF2340/v05/foiler/sim04.pdf Now imagine a string under tension T and of mass density rho confined to the surface of a cylinder of radius r. I posit that this string will act just as...
  25. A

    How to find a trivialization of a torus

    I am confused with trivialization of a tangent bundle. Can anyone can help me solve the problem of finding a trivialization of the tangent bundle of the torus S^1*S^1 in R^4. Thanks.
  26. B

    Prove volume of a torus equation

    how does the formula for volume of a torus work. is there a proof with integration?? could you use an ellipse?
  27. D

    Maping from R2 to R3 torus, finding max min and saddle points

    Homework Statement this is from ch9 (functions of several variables)of baby rudin a,b real b>a>0 define a mapping f=(f1,f2),f3) of R2 into R3 by f1(s,t)=(b+acos(s))cos(t) f2(s,t)=(b+acos(s))sin(t) f2(s,t)=asin(s) I showed that there are exactly 4 points p in K=image(f) such that...
  28. R

    How do I calculate the moment of inertia of a torus with given parameters?

    Homework Statement Find the moment of inertia of a torus if mass is m and density \rho is constant. The cross-sectional radius is 'a' and the radius is R. Homework Equations I= \int r^2 dm The Attempt at a Solution Well I looked up the answer to be I_z= m(R^2 +...
  29. K

    Calc 2: Volume of a Torus Homework - Solve Step-by-Step

    Homework Statement http://faculty.tcu.edu/richardson/Calc2/H20090323torusVolume.htm This is a link to the homework. 2. The attempt at a solution I did number 1 by doing 2 * integral from -2 to 2 of sqrt(4-x^2) = 4pi The second problem is where I am completely confused and don't know...
  30. quasar987

    Sphere, torus, degree, cohomology

    I am trying to show that any smooth map F from the 2-sphere to the 2-torus has degree zero. The definition of the degree of F can be taken to be the (integer) degF such that \int_{\mathbb{S}^2}F^*\omega=(\deg F)\int_{\mathbb{T}^2}\omega For omega any 2-form on T². Another definition would be...
  31. R

    Differential equation in the torus

    Hello all. Consider the torus T^2 as a subset of R^3, for example the inverse image of 0 by the function f(x,y,z)=(\sqrt{x^2+y^2}-1)^2+z^2-4. I need to obtain a example of a vector field X defined in the whole R^3, such that: 1) X is invariant in the torus 2) the orbits of X in the torus are...
  32. S

    Finding The Volume Enclosed by a Torus

    Homework Statement Find the volume enclosed by the torus rho = sin theta. Homework Equations The Attempt at a Solution I tried setting the limits as phi from 0 to pi, theta from 0 to 2 pi, and rho from 0 to sin theta. However, if i do that, i get a volume of 0. How should i...
  33. E

    Help with a special kind of Torus

    Hi, Consider a cone with height H and radius of base circle R. Take a point on the circular edge of the cone and make that the center of another circle of radius r whose normal points at the apex of the cylinder. Rotate this circle around the axis of the cone to create a surface. Given an...
  34. A

    Vector Field Describing Fluid Flow in a Torus

    Homework Statement Write a vector field equation which describes fluid flowing around a pipe of radius r whose axis is a circle of radius R in the (x,y)-plane. Homework Equations x2+y2=r2 Equation of a torus? The Attempt at a Solution What I've gathered from the question: the pipe...
  35. M

    What is the optimal shape for a floatation device under high external pressure?

    Right now I'm trying to design a floatation device for putting 1000m under the sea (~1500 psi external pressure). The nicest looking and most convenient shape would be a torus but I'm having trouble estimating stress on the thing. I'm looking for around 20lb of buoyancy or more and dimension are...
  36. A

    Form of line element of a torus.

    I noticed somewhere the line element of a two-dimensional torus is written in the form ds^2=r^2(d\theta^2_1+d\theta^2_2) The author only states that he assumes same radius parameter for simplicity and no further explanation is given. But I do not understand how that form is possible. I...
  37. stevebd1

    Gravitational acceleration for a rotating torus

    gravity of a rotating torus While Gm/r^2 seems to apply only for gravitational acceleration towards spheroids, what would be the equation for gravitational acceleration towards a rotating torus? I'm sure the equation would be the same for spheroids and toroids at large distances but what would...
  38. C

    Find All Stresses on a Torus Pressure Vessel

    I have been goven the task of finding all the stresses on a Torus (i.e. donut) Pressure vessel. Any help would be much appreciated!
  39. R

    How do I find the line element on a torus?

    Hey folks, Does anyone know how to get the line element on a torus? The ds^2 term. I want to find the metric for toroidal geometry. Any help appreciated! Richard
  40. S

    Air Torus Hoverboard: Is it Possible?

    Hi everyone. As you may know, it is possible to create a torus or toroidal vortex of air. Some toys such as the air zooka create a torus of air that travels across a room to knock down a light object. Do you think it is possible to create a device that produces a continuous torus of air...
  41. B

    Arc Length around Helical Torus

    Can anybody help? Mathematical Physics. I'm seeking an analytical expression for the path length of a point that follows a helical path with the helix wound about an axis to form a torus. The arc path length of a helix is simple to compute, but when its formed into a torus there is a...
  42. N

    Mapping torus is an (m+1)-manifold

    Homework Statement Let X be an m-manifold. Let M(f) be the space obtained from X\times [0,1] by gluing the ends together using (x,0)\sim (f(x),1). Show that if M is an m-manifold then M(f) is an (m+1)-manifold. The Attempt at a Solution Since X has an atlas \{ (U_\alpha,\varphi_\alpha) \}...
  43. P

    Boundary of a torus or sphere?

    Homework Statement What is the boundary of a torus? What is the boundary of a sphere? The Attempt at a Solution Both 0?
  44. G

    What is the distance between 2 points on a torus?

    I was playing with a 2D model with point particles and gravitational attraction. To avoid particles going off to infinity, I thought I might turn the 2D space into a topological torus: connect x=L = x=0 and y=L to y=0. But now a problem occurs: What is the distance between 2 points? Because of...
  45. P

    Torus, two graphs and their equivalency

    Homework Statement Two graphs defined for a two dimensional torus: f_1(t) = (\frac{1}{\sqrt{2}}(a\ +\ b\ sin\ t),\frac{1}{\sqrt{2}}(a\ +\ b\ sin\ t),b\ cos\ t),\ t \in (-\frac{\pi}{2},\frac{\pi}{2}) f_2(t) = (a\ cos(t+\frac{\pi}{4}),a\ sin(t+\frac{\pi}{4}),b),\ t \in...
  46. D

    Net electric flux through torus

    Homework Statement What is the net electric flux through the torus (i.e., doughnut shape) of the figure Homework Equations net flux= E*A I believe is needed The Attempt at a Solution I don't know how to do this problem at all. I feel like I don't have enough information to...
  47. T

    Shell's Method: Solids of Rotation, Torus

    Homework Statement Volume of Torus: using Shell's mehod 4\pi \int^{1}_{-1}((R-x) \sqrt{r^2 - x^2})dx Homework Equations The Attempt at a Solution I don't know how to integrate this at all. I cannot use any conventional methods...or I can't think of a way... i.e. use isolate a function as...
  48. P

    AGNs, and why there is a torus

    Hello, In one of my lectures that I had a few weeks ago, the lecturer was talking about Active galactic nuclei, he spoke of a model in which there was a BH at the centre of a galaxy, and that around the BH, there is a torus of gas and dust, and so it explains the seyfarts 1 & 2 which are simply...
  49. M

    Tangent Plane for a torus

    I know I am making a stupid mistake but I am not sure what it is... Find an equation for the plane tangent to the torus X(s,t)=((5+2cost)coss, (5+2cost)sins, 2sint) at the point ((5-(3)^1/2)/(2)^1/2, (5-(3)^1/2/(2)^1/2, 1). First I have to find what s and t are in order to sub them in for...
  50. S

    What is the process for calculating stress on a torus in a high pressure vessel?

    I am a novice at stress analysis. For my competition, I need to calculate the stress on a torus in vacuum filled with air at pressure P. Here's is what I have managed to do. I have attached an image below. The torus section is an angular section of a curved cylinder. Its angular dimensions are...
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