- published: 25 Apr 2014
- views: 167685
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis 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.
Real-world examples of (approximately) toroidal objects include inner tubes, swim rings, and the surface of a doughnut or bagel.
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 approximations include doughnuts, vadai or vada, many lifebuoys, and O-rings.
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 three-dimensional Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane. This produces a geometric object called the Clifford torus, a surface in 4-space.
Brush off my shoulders, but not 'cause it's colder
Will it get better, now that we're older?
Remember the feeling, a tear of emotion
To be high as a mountain, as deep as the ocean
How I wish I could turn back time (x11)
Oh, I just hate being without you
It's just another thing to break through
Carry the hurt I dare not speak it
Bring back the love that we all believe in
How I wish I could turn back time (x11)