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).
hi,
for most of you this might be a simple question:
Is it possible to embed the flat torus in Euclidean space?
If we, for example, take a rectangle and identify the left and the right sides we get a cylinder shell, that can be embedded easily in R^3. If we construct the...
Hi, I need to find the volume of the solid that lies above the cone with equation (in spherical coordinates) \phi = \frac{\Pi}{3} and inside the torus with equation \rho = 4\sin\phi . I thought that the bounds are: 0\leq\rho\leq4\sin\phi, \frac{\Pi}{3}\leq\phi\leq\frac{\Pi}{2}, and...
I'm asked to consider regular networks on a torus. I'm given that V - E + F = 0. I need to show it is impossible to have a regular network on a torus where the faces are pentagons; I don't understand that at all. Surely it is easy to have pentagons as faces… All you would need to is draw a...
Hi!
This may not be the right place for it but I have a question about the torus.
In the centre point, the exact middle of the hole in the torus if a let's say, a perfect sphere was placed there, would it simply stay in the one place if everything was stationary?
Also could you walk all...
Hi,
A small but exceptionally annoying algebraic topology question:
I'm trying to find the Hodge numbers (from the Hodge-de Rham cohomology) for a 2n-dimensional torus (that is, n complex dimensions).
Anyone have any ideas? It's a rather technical question, but I don't really want to...
There is a developing theory that the Universe may be shaped like a three-torus, the mathematical equivalent of a rubber cube that's bent so that all opposing sides are connected. This would mean that the Universe is finite, but does not have the problematic edge that's included in most...