500 torus

a line segment” is valid for the entire class of objects

topologically equivalent to a circle.

Any transformation that requires puncturing or

tearing a surface, or joining together two disjoint por-

tions of a figure, is not considered a valid topological

action. (Tearing a surface, for instance, separates points

that were close together.) As it is impossible to deform

a

SPHERE

into a

TORUS

without puncturing the surface,

these two shapes are not topologically equivalent. As

other examples, the capital letters C, M, and Z are

topologically equivalent, as are the letters D, O, P, and

R, but no letter from the first group is topologically

equivalent to a letter in the second. No letter in the

alphabet is topologically equivalent to the letter B

(other than capital B itself).

The study of topology began with L

EONHARD

E

ULER

(1707–83) and his analysis of the famous

SEVEN BRIDGES OF

K

ÖNIGSBERG PROBLEM

. The mathe-

matician C

ARL

F

RIEDRICH

G

AUSS

(1777–1855) exam-

ined the distortion of knots and invariant properties

that arise in the study of

PROJECTIVE GEOMETRY

. In

1895 French mathematician H

ENRI

P

OINCARÉ

(1854–1912) examined these works and published

five papers, laying a theoretical framework for a for-

mal study of topology.

The discipline of point-set topology attempts to

capture the notion of “closeness” without making

mention of distance. In this theoretical approach, one

is given a set Xand a specified collection of subsets

called open sets, which, in some loose sense, are meant

to represent those points that are deemed “close” to

each other. These subsets are required to satisfy three

basic properties:

1. The empty set Ø and the entire set Xare deemed

open.

2. If Aand Bare two open sets, then their intersection

A∩Bis also an open set.

3. The union of any collection of open sets is another

open set.

Any set Xwith a collection of subsets classified as open

satisfying the three stated properties is called a topolog-

ical space. For example, the set of real numbers with

unions of open

INTERVAL

s as the open sets is a topolog-

ical space. So too is the set X= {a,b,c,d} with subsets

Ø, {a}, {b,c,d}, and Xas the open sets. Two topological

spaces are said to be equivalent if there is a one-to-one

correspondence between the points in one space and

the points in the other so that open sets in one space

correspond to open sets in the other.

See also

KNOT THEORY

; M

ÖBIUS BAND

.

torus (anchor ring, donut) A closed curved surface

possessing a single hole akin to the surface of a donut

or an inner tube is called a torus (plural: tori). A torus

can be regarded as a

SOLID OF REVOLUTION

obtained

by rotating a circle in a plane one full revolution about

a line that does not intersect the circle. (This circle is

called the generating circle.) As such, P

APPUS

’

S THEO

-

REMS

show that the

VOLUME

Vand the surface

AREA

A

of a torus are given by the formulae:

V= 2π2a2b

A= 4π2ab

where ais the radius of the circle rotated about a line,

and bis the distance of the center of the the circle from

that line. These formulae can also be obtained by the

techniques of

INTEGRAL CALCULUS

In a C

ARTESIAN COORDINATE

system, if the circle of

radius ais positioned in the yz-plane with center at

position balong the y-axis and is rotated about the z-

axis, then the equation of the resulting torus produced

is given by:

This follows from P

YTHAGORAS

’

S THEOREM

applied to a

right triangle within the generating circle when it is

rotated to a position so that the x-, y-, and z-coordinates

of a point on the torus are (x,y,z).(Examine the right tri-

angle of height zwith radius aas its hypotenuse.)

If θdenotes the angle of a point on the generating

circle, and ϕthe angle through which that generating

circle is rotated about the z-axis, then one obtains the

following

PARAMETRIC EQUATIONS

for the torus:

x= (b+ acosθ)cos ϕ

y= (b+ acosθ)sin ϕ

z= asinθ

Each point on the generating circle of a torus, when

rotated about the z-axis, traces out a circle. Thus a