Torus

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Torus

This picture shows a torus formed by rotating a circle around the z-axis


Contents

Basic Description

A torus is a surface formed by revolving a circle about a line. Often, tori (the plural of "torus) are doughnut shaped, as in the main image here. It is formed by revolving a circle about the z-axis. It can also be described as a ring-shaped surface with a hole. Another way of picturing the torus is to imagine an open-ended thick tube whose ends have been glued together. The solid contained by the surface is called a toroid. Tori are commonly used for illustrating concepts in geometry and other topics, as they are simple figures with very many interesting properties.

A More Mathematical Explanation

Mathematically, we can describe tori using several different equations.

For the sake of simplicity, [...]

Mathematically, we can describe tori using several different equations.

For the sake of simplicity, we'll assume that our torus is created by revolving a circle around the z-axis, as in the main image.

Let the radius from the center of the hole to the center of the torus tube be R, and the radius of the tube be r, as shown below in a cross section cut away. This cross section is a slice in the z direction--meaning that we're looking at the cut away from the side of the torus. Up corresponds to the positive z direction. The center of the torus is marked with a blue dot, the the cross section is a blue circle.

We can also see these values by looking down on torus from above. Again the center of the torus is a blue dot, and the torus is in between the purple and blue circles. This is a slice along the z=0 plane.

In Cartesian coordinates, all of the points on a torus along the z axis centered at the origin (as in the main image) satisfy the following equation:

\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2, \,\!

The parametric equations that describe this type of torus are:

x(u, v) =  (R + r \cos{v}) \cos{u} \, y(u, v) =  (R + r \cos{v}) \sin{u} \, z(u, v) =  r \sin{v} \,

where u and v both range between 0 and  2\pi .

Types of Tori

There are three possible types of tori depending on the relative sizes of R and r. If R>r, it is a ring torus, as shown in the main picture. This is the typical doughnut shape. If R=r, then it is a horn torus. A horn torus is tangent to itself at the Cartesian coordinate 0,  0,  0. If R<r, then it becomes a self-intersecting spindle torus. Collectively, these types of tori are known as the standard tori. If no specification is made as to which type the term torus refers to, it usually is taken to mean the ring torus. The images below show different views of each type of torus.





















Surface Area and Volume of a Ring Torus

For ring tori and horn tori, the surface area and volume are easy to define, since there is obviously an inside and an outside to the shapes.

We can find the surface area and volume of these tori by thinking of them as cylinders wrapped around the origin. First note that, when viewed from above, the torus has inner and outer,  r_{in}=R-r and  r_{out}=R+r respectively. These give us an inner circumference C_{in}=2 \pi (R-r) and  C_{out}=2 \pi (R+r) .

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To find the area and volume, we want to cut open the torus and treat it like a cylinder. But, when the torus is cut and unwrapped, we can see that it is a different shape that has a length  C_{in} on one side and  C_{out} on the other, as shown below. The radius of this tube is still  r .



Fortunately, we can treat this as a cylinder with a length that is the average of  C_{in} and  C_{out} . This is due to the symmetry of the torus.



Picture this by imagining cutting off the top of the pointed cylinder-like shape (shown in blue to the left), flipping it over and placing it on the lower side of the cylinder-like shape (also shown in blue). This gives us our cylinder with length  C_{ave}


L= C_{ave}=\frac{C_{in}+C_{out}}{2}=\frac{2 \pi (R-r)+2\pi (R+r)}{2}=2 \pi R

Now that we have a formula for the length of a cylinder, and we also know it's radius. That means we can find the surface area and volume using standard formulas for cylinders:

 Area= 2 \pi r L= 2\pi r (2\pi R)=4 \pi^2 rR
 Volume= (\pi r^2)L=\pi r^2 (2 \pi R)=2 \pi^2 r^2 R .



n-torus

There are two definitions of the term n-torus. The first relates to the number of holes a torus can have. A torus can have multiple holes. A torus with one hole is simply referred to as "the torus". A torus with two holes is called the 2-torus or the double torus, one with three holes is the 3-torus or triple torus. A torus with n holes is the n-torus.

The other definition of n-tori is in relation to the number of dimensions. In one dimension, a line bends into a circle, so a torus in 1D is a circle. In two dimensions, a rectangle folds to form a tube, whose ends when connected form a 2-torus, the usual kind of torus. In 3D, a cube wraps to form a 3-torus. In general, n-torus objects exist in n+1 dimensions







Coloring a Torus

Given a torus divided in to regions, we can color each region so that adjacent regions always have different colors. Whereas on the plane or sphere at most 4 colors are needed to color any such division into regions, on the torus there exist divisions into regions that require 7 colors! One such division is shown below. More than 7 colors are never necessary. This Seven Color Theorem for the Torus is actually much easier to prove than the 4-color theorem for the plane, and was proved much earlier.

Image:Torus.png

For Tori in higher dimensions, see Projection of a Torus.


Why It's Interesting

Additionally, a theorem exists which details how any closed surface "is either homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes"[1][2]. Follow this link for an explanation: Classification Theorem for Compact Surfaces. This makes the torus a very fundamental and versatile manifold.


Teaching Materials

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References

  1. Massey, William. (1991). A Basic Course in Algebraic Topology (Graduate Texts in Mathematics). New York: Springer-Verlag.
  2. Surface. May 27, 2011 . Wikipedia, The Free Encyclopedia. ttp://en.wikipedia.org/w/index.php?title=Surface&oldid=431219154. Accessed: July 5, 2011.





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