Surface of genus one that looks like this:

        ..----..
     .-          -.
   -                -
  |     .------.     |
   .   --______--   .
    .              .
     .__        __.
        ---__---
                          

Can be created by identifying

  --------------
 |      B       |
 |              |
 |A            A|
 |              |
 |      B       |
  --------------    
the edges with the same label.

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.

To"rus (?), n.; pl. Tori (#). [L., a round, swelling, or bulging place, an elevation. Cf. 3d Tore.]

1. Arch.

A lage molding used in the bases of columns. Its profile is semicircular. See Illust. of Molding.

Brande&C.

2. Zool.

One of the ventral parapodia of tubicolous annelids. It usually has the form of an oblong thickening or elevation of the integument with rows of uncini or hooks along the center. See Illust. under Tubicolae.

3. Bot.

The receptacle, or part of the flower on which the carpels stand.

4. Geom.

See 3d Tore, 2.

 

© Webster 1913.

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