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. A

    I Foliation of the 3-torus

    I'm trying to understand more about the Hopf map and, I think I can see that the torus knot K1 defines the boundary of a looped, twisted ribbon embedded in the interior, aka the Mobius strip. So slicing the torus open along the knot boundary means you have two halves of the torus linked...
  2. Mastermind01

    A Hodge decomposition of a 1-form on a torus

    I was reading Dunne's review paper on Chern-Simons theory (Les-Houches School 1998) and I don't get how he decomposes the gauge potential on the torus. My own knowledge of differential geometry is sketchy. I do know that the Hodge decomposition theorem states that a differential form can be...
  3. H

    What is the symbol of this torus?

    Hi Pfs, I can get a taurus from a square: I identify the oppsite sides. It has the symbol $R^2/ Z^2 $ Suppose now that i replace this square by a rectangle with the length L ans 2L. I identify the opposite sides in the same manner. The new taurus is also a quotient of $R^2$ but how to write it?
  4. H

    Equator of a Planet with Four Spatial Dimensions is a 4D Torus

    A 4D planet has no axis of rotation. Nothing special about planets, all freely rotating 4D bodies have two perpendicular planes of rigid rotation. (Clifford proved this in the 19th century.) Now there is nothing stopping us from thinking of planes of rotation here in 3D Universe. It's the...
  5. ergospherical

    I Transform Coordinates for Torus Metric in Wald

    I can't figure out how to transform the coordinates to get to the given metric \begin{align*}ds^2 = \cos x (dy^2 - dx^2) + 2\sin x dx dy \end{align*} for a 2-torus. Initially I parameterised it by two angles ##\theta## (around the ##z## axis) and ##\phi## (around the torus axis), to write...
  6. A

    I Time Dilation on 3D Torus: Clock Speed & Why It Matters

    Assume that space is a three-dimensional torus ( a 3D donut) . If there is a clock traveling at a CONSTANT speed in a direction parallel to the torus (inside out of the hole) and one clock that is still. Which clock ticks faster and why? I know that the clock rotating will tick slower, but I...
  7. Spinnor

    I Hopf Fibration, torus with fibers, do it 2 ways?

    The following is a projection of a set of fibers of a 3-sphere? Consider the mirror image of the above, Can the 3-sphere be rotated so that (b) becomes (d)? Is there an easy way to understand this? Thank you.
  8. docnet

    I How is this a representation of a 3 dimensional torus?

    In a differential geometry text, a torus is defined by the pair of equations: I initially thought this was somehow a torus embedded in 4 dimensions, but I do not see how we can visualize two orthogonal 2-dimensional Euclidian spaces. How is this a representation of a 2 dimensional torus...
  9. V

    A The map from a complex torus to the projective algebraic curve

    I am following the proof to show that the complex torus is the same as the projective algebraic curve. First we consider the complex torus minus a point, punctured torus, and show there is a biholomorphic map or holomorphic isomorphism with the affine algebraic curve in ##\mathbb{C}^2##...
  10. A

    Doubt about a unit vector in toroidal coordinates

    The system considers a torus that has a wire wrapped around it, through which a current flows. In this way, a field originates in the phi direction. The direction of current is "theta" in the spherical coordinate system but in toroidal system, in several book shows that the electrical current...
  11. Alettix

    Magnetic field in the gap of a tapered torus

    Homework Statement Consider a toroidal electromagnet filled with a magnetic material of large permeability µ. The torus contains a small vacuum gap of length h. Over most of its length the torus has a circular cross section of radius R, but towards the gap the torus is tapered on both of its...
  12. Ryan Doucette

    Nuclear Fusion: Electron Thermal Transport Terminology

    I am an undergrad physics major in my final semester currently taking Intro to Thermodynamics. As a final project, each student must choose a topic related to thermodynamics that is more advanced than what is covered in the curriculum and write a paper and present our findings to the class on...
  13. A

    I Integrating a Function over a Torus Contour

    I'd like to integrate a function over a closed circle-like contour around an arbitrary point on a torus and I assume I would use the expression: $$ \int_{t_1}^{t_2} f(x,y,z) \sqrt{x'(t)^2+y'(t)^2+z'(t)^2}dt$$ And I cannot come up with an explicit parameterization of the variables in terms of...
  14. T

    I Surface area of inner and outer halves of torus

    A torus with major radius, ##R##, and minor radius, ##r##, has a total surface area given by ##4\pi^2 Rr##. If one slices the torus on its midline (i.e. at a line on a poloidal angle of ##-\pi/2## and ##\pi/2##), I was told the inner half of the torus has a smaller surface area than the outer...
  15. PsychonautQQ

    A Torus 2-sheet covering of Klein bottle

    So I'm trying to understand how the Torus is a 2-sheet covering of the Klein bottle. I found this on math exchange: https://math.stackexchange.com/questions/1073425/two-sheeted-covering-of-the-klein-bottle-by-the-torus. The top response add's rigor to of the OP's observation that the double...
  16. L

    A Can I find a smooth vector field on the patches of a torus?

    I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...
  17. A

    I What is an atlas for a torus manifold?

    I don't quite understand some work I'm doing creating the normal Riemann surface for the function ##f(z)=\frac{A}{\sqrt{(1-z^2)(k^2-z^2)}}##. I can use Schwarz-Christoffel transforms to map the function to a rectangular polygon in the zeta-plane then map this rectangle onto a torus. But I...
  18. FallenApple

    A Detecting a Torus in a data cloud

    Would the following method work? I could uniformly distribute points into the data cloud. Of the "darts" that I threw in, create edges between all points with a metric value under a certain amount. The nodes in the resulting graph that have more neighbors would indicate greater density. I could...
  19. A

    I Contort rectangle to torus minus splice=homomorphism?

    I understand the transformation in general is not homomorphic but what about the transformation minus the splices, that is, contort it all the way up to and not including splicing the edges? Isn't that a homomorphism? Can't we define a bijective function (rotation matrices) to map the two...
  20. Wendel

    Constructing a Torus: Logical Justification

    Homework Statement Suppose we wish to construct a torus {(x,y,z)∈ℝ³:(R-√(x²+y²))²+z²=r²} whose axis of symmetry is the z-axis and the distance from the center of the tube to the center of the torus is R. Let r be the radius of the tube. Homework Equations The large circle in the xy-plane...
  21. G

    B Preferred directions on a torus?

    I would like to regard an intrinsically flat 2-torus. This is usually sketched as a square with the left and right edge identified and the upper and lower edge identified, respectively. The four corners of the square represent the same point. Now I connect this point to itself via a loop on the...
  22. A

    Mathematica Mathematica: Mapping a Four Color Rectangle to a Four Color Torus

    Hi, I was wondering if someone could help me understand the grid command in the following Mathematica code that transforms a rectangle to a torus like in this video below. My problem is I want the blue rectangle in the Mathematica code to look like the four color rectangle in the plot below...
  23. L

    A About Chern number of U(1) principal bundle on a torus

    The U(1) bundle on a torus is a important math setup for a lot of physics problems. Since I am awkward on this subject and many of the physics material doesn't give a good introduction. I like to put some of my understanding here and please help me to check whether they are right or wrong. 1. A...
  24. L

    A Stokes' Theorem and Curvature on a Torus

    I am now looking at a physics problem that should be a use of stokes' theorem on a torus. The picture (b) here is a torus that the upper and bottom sides are identified as the same, so are the left and right sides. ##A## is a 1-form and ##F = dA## is the corresponding curvature. As is shown in...
  25. A

    I Mapping double cover over a torus

    I am interested in plotting contours (and integrals) over the algebraic function ##w^2-(1-z^2)(1-k^2 z^2)=0## on it's normal Riemann surface, a torus. Anyone here interested in helping me with this? I have the basic idea just the details I'm having problems with. Would be a nice educational...
  26. Jovy

    How Do You Calculate the Volume of a Torus Formed by Revolving a Circle?

    Homework Statement A torus is formed by revolving the region bounded by the circle ##x^2+ y^2= 1## about the line x =2 Find the volume of this “doughnut-shaped” solid. (Hint: The integral ##{\int_{-1}^1} \sqrt{1 -x^2} dx## represents the area of a semicircle.) 2. Homework Equations...
  27. peroAlex

    Oscillation of a Charged Particle

    Homework Statement At our university we were given this problem: charged ball with mass of ##m = 0.0001 kg## and charge ##Q = -10^{-5} C## is placed on geometric axis of thin torus with inner radius of ##r_{inner} = 0.05 m##, outer radius of ##r_{outer} = 0.1 m## and surface charge density...
  28. P

    MHB Mass of a Torus in Spherical Coordinates

    consider a torus whose equation in terms of spherical coordinates(r,\theta,\phi) is r=2sin\phi for 0\le\phi\le2\Pi. determine the mass of the region bounded by the torus if the density is given by \rho=\phi.
  29. S

    Gravitational force on an astronaut from a nearby massive torus

    Homework Statement There are a big object and an astronaut in space. How do we calculate the gravitational force between them. I enclose a photo. I have given the mass of an astronaut, the dimensions of this giant ring and density of the ring. There is also a mistake in the photo. The astronaut...
  30. dkotschessaa

    A Is This the Correct Way to Attach Möbius Strips to a Torus?

    Another zany homology question: Just let me know if I have my labeling scheme right so far. I have a torus. I have cut two holes into it and am attaching Möbius strips around the holes. (Clearly we are not in 3 dimensions?) My Torus is represented by a polygon with the labeling scheme ## a...
  31. zinq

    A Can classical gravity be defined on a torus?

    Some time ago I tried to define classical inverse-square gravity on a 3-dimensional (cubical) torus T3: the quotient space obtained by identifying opposite faces of a unit cube. (Or more rigorously, the quotient space T3 = R3/Z3 of R3 by its subgroup Z3 of integer points.) I assumed there...
  32. Tom MS

    I Can a Torus be Gravitationally Bound?

    Now I asked a question the other day about the gravitational binding energy of a torus, and someone responded that it cannot be gravitationally bound purely, but requires some opposing force. Okay, fine. But, qualitatively, can a toroidal planet be gravitationally bound if it has another force...
  33. Tom MS

    I Gravitational Binding Energy of a Torus

    I was looking at the wikipedia page for the gravitational binding energy of a sphere, but let's say that there was a toroidal planet. What would its gravitational binding energy be? I have attempted the solution similar to what they did on wikipedia and obtained: U = -4 G \pi^5 R^2 r^4 \rho^2...
  34. T

    A Why is the Brillouin zone a torus?

    Recently, topological concepts are popular in solid state physics, and berry connection and berry curvature are introduced in band theory. The integration of berry curvature, i.e. chern number, is quantized because Brillouin zone is a torus. However, I cannot justify the argument that...
  35. Q

    Comparing V and SA of a torus vs sphere

    Homework Statement A torus has a major radius and a minor radius. When R>r by a magnitude of at least 4x, it comes to be a slim ring looking shape. When R>r by a magnitude of 1/2, the shape looks to be a donut. When R=r, the torus shape looks more like a sphere except with a small gap in the...
  36. Mr. Rho

    Mathematica Rotation of 3D Plot using Euler angles

    So, I'm trying to plot a 3D "dipole" (an arrow with a small torus around it basically) in mathematica, and want to rotate it according to Euler angles... I use this code for the rotation matrix: rot[a, b, g] := RotationMatrix[g, {1, 0, 0}].RotationMatrix[b, {0, 1, 0}].RotationMatrix[a, {0, 0...
  37. Jonathan Swift

    Does a Torus dynamo actually work?

    Hi all, in short does a Torus shaped generator actually generate any useful power? If so, is there any formula/calculator that can predict its output in watts? I've seen some claims online but I'm skeptical. They are basically like a shake torch with a solenoid mechanism in a torus shaped...
  38. A

    Gravity on a 1-D Torus: Understanding the Conceptual Simplification

    I am trying to make understand gravity on a flat 2-D torus. To help myself get my head around it, I simplify the problem into a 1-D torus. Let's have a 1-D space (or line). Instead of the left part of the line going infinitely to the left and the right part of the line going infinitely to the...
  39. M

    Can Penrose Stairs Exist on a Torus? Exploring the Possibilities

    Nobody mentions that Penrose stairs is possible on the torus. I wonder Why. isn't this obvious?
  40. Z

    Flat Universe as Torus: Legit Research & Solutions?

    A lot of cranks on the net make a lot of the torus - of course magnetic fields are toroidal - but not space time right? Or could it be - a torus is flat (zero gaussian curvature) which is how we observe the universe to be as far as we have measured it's curvature on large scales - and it would...
  41. H

    Torus used to model 4D rotation

    A torus can be used to model rotations of a sphere in 4 dimensions. Such rotations have two planes of rotation at right angles to one another. So one rotation plane corresponds to rotation around the major axis of the torus, and the other rotation plane to rotation around the minor axis...
  42. P

    T-Duality on Torus: Inverting Radii & Interchanging A- & B-Cycles

    I think I have a relatively decent grasp on T-duality where we've compactified S^{1} . However, when compactifying a 2-torus, is the T-duality transformation where you invert both radii of the two circles simultaneously, or is the claim that you can invert one of the two, leaving the other...
  43. Mr. Rho

    Poloidal current in toroidal solenoid

    Hi, I'm trying to figure out how the current density for a poloidal current in toroidal solenoid is written. I found you may define a torus by an upper conical ring ##(a<r<b,\theta=\theta_1,\phi)##, a lower conical ring ##(a<r<b,\theta=\theta_2,\phi)##, an inner spherical ring...
  44. arivero

    Is orbifolding of the torus a sophisticated spherification?

    I was thinking... perhaps are all these orbifolds with conical singularities simply ironed versions of the typical manifolds of Kaluza Klein? I asked in math overflow about how to get spheres and CPn spaces from n-dimensional torus and the people who answered seem to look at it as usual...
  45. Jimster41

    Question about a flat torus topology

    I'm trying to get comfortable with the idea of a flat torus topology that is also an everywhere a smooth manifold like the video game screen where you got off the screen to the right and pop out on the left (because as I understand it this topology could be a model of space) I can't get how...
  46. J

    Torus parametrization and inverse

    I've been looking at the torus parametrization \begin{equation} \phi(u,v)=((r\cos u+a)\cos v, (r\cos u +a)\sin v, r\sin u) \end{equation} with \begin{equation}a>0, r\in(0,a)\end{equation}. I want to invert this map to get a chart map for the torus. Can anyone give me a hand with this? Thanks!
  47. Philip

    Regarding the complex solutions of a torus

    The equation for a torus defined implicitly is, $$(\sqrt{x^{2} + y^{2}} -a)^{2} + z^{2} = b^{2}$$ When solving for the z-axis in the torus equation, we get complex solutions, from the empty intersection: $$z = - \sqrt{b^{2} - a^{2}}$$ $$z = \sqrt{b^{2} - a^{2}}$$ I was told by someone that...
  48. TESL@

    Find a Machinable Plastic/Glass Torus - R=15cm, r=8cm

    Hello, Do you know any online source where I can purchase a plastic or glass torus which can be machined easily? The requirements are low conductivity and high temperature resistance. (I know it is unlikely for plastics, but worth asking) Thank you. edit: Forgot to tell dimensions: R = 15 cm...
  49. Math Amateur

    Torus - Singh, Example 2.2.5 - Baffled by certain aspects

    I am baffled by some aspects of the torus ... I hope someone can help ... I am puzzled by some aspects of Singh's treatment of the torus in Example 2.2.5 ( Tej Bahadur Singh: Elements of Topology, CRC Press, 2013) ... ... Singh's Example 2.2.5 reads as follows: My questions related to the...
  50. TESL@

    Parametrizing a Complex Curve on a Torus Surface

    Hello, I am currently trying to parametrize a surface constructed by thickening a rather complicated curve, defining its normal, binormal and tangent vectors. Even using Mathematica simplification, the resulting vectors are page long expressions and the reason for it is because I have four...