torus (thing)

Despite the approximation given by the ASCII art above, most mathematicians can identify with a torus most easily as the shape of a ring donut. A torus has two key measurements when defining: an inner radius R which measures the distance from the centre to the middle of the actual ring, and an outer radius r for the distance from the inside of the ring to the outside. To continue the donut analogy, R corresponds to the size of the hole (almost), but r measures the thickness of the donut. Cut through view from the side:

  ____           ____
 /    \         /    \
|      |       |      |
 \____/         \____/
   
   ^-------^       ^--^
      R              r

Compare this to a sphere, which has only one defining characteristic, namely its radius.

Note that the donut is a specifically three dimensional torus, but a torus is equally well defined in higher dimensions and is frequently encountered in 4D geometry. The 3 dimensional torus is occasionally known as an anchor ring.

The surface area of such a torus can be found by:
S = 4π²Rr

The volume of a torus is:
V = 2π²Rr²

The Cartesian equation for a three-dimensional torus with its symmetry about the z-axis is

[ R - √(x² - y²) ]² + z² = r²

Thanks to Mathworld, we can also specify the parametric equations for such a torus, as:

x = (c + a·cos(v) ) · cos(u)
y = (c + a·cos(v) ) · sin(u)
z = a·sin(v)

for u,v ε [0, 2π).

There are three types of torus:
Ring torus: This is the typical torus, when r < R.
Horn torus: When r = R, so the torus is tangent to itself at the centre.
Spindle torus: A torus that intersects itself, as r > R.

The torus is a surface of genus 1, which (for the layman) means it has one hole through it. By contrast, a sphere has genus 0. For more information on such characteristics, read up on topology.