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 portions 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 LEONHARD
EULER (1707–83) and his analysis of the famous
SEVEN BRIDGES OF KÖNIGSBERG PROBLEM. The mathematician CARL FRIEDRICH GAUSS (1777–1855) examined the distortion of knots and invariant properties
that arise in the study of PROJECTIVE GEOMETRY. In
1895 French mathematician HENRI POINCARÉ
(1854–1912) examined these works and published
five papers, laying a theoretical framework for a formal 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 X and 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 X are deemed
open.
2. If A and B are two open sets, then their intersection
A ∩ B is also an open set.
3. The union of any collection of open sets is another
open set.
Any set X with a collection of subsets classified as open
satisfying the three stated properties is called a topological space. For example, the set of real numbers with
unions of open INTERVALs as the open sets is a topological space. So too is the set X = {a,b,c,d} with subsets
Ø, {a}, {b,c,d}, and X as 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; MÖBIUS 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, PAPPUS’S THEOREMS show that the VOLUME V and the surface AREA A
of a torus are given by the formulae:
V = 2π
2
a
2
b
A = 4π
2
ab
where a is the radius of the circle rotated about a line,
and b is 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 CARTESIAN COORDINATE system, if the circle of
radius a is positioned in the yz-plane with center at
position b along the y-axis and is rotated about the zaxis, then the equation of the resulting torus produced
is given by:
This follows from PYTHAGORAS’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 triangle of height z with radius a as its hypotenuse.)
If θ denotes the angle of a point on the generating
circle, and ϕ the 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
x y
b
z
a
2
2
2
2
2
+
−
+
=
500 torus
topologically equivalent to a circle.
Any transformation that requires puncturing or
tearing a surface, or joining together two disjoint portions 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 LEONHARD
EULER (1707–83) and his analysis of the famous
SEVEN BRIDGES OF KÖNIGSBERG PROBLEM. The mathematician CARL FRIEDRICH GAUSS (1777–1855) examined the distortion of knots and invariant properties
that arise in the study of PROJECTIVE GEOMETRY. In
1895 French mathematician HENRI POINCARÉ
(1854–1912) examined these works and published
five papers, laying a theoretical framework for a formal 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 X and 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 X are deemed
open.
2. If A and B are two open sets, then their intersection
A ∩ B is also an open set.
3. The union of any collection of open sets is another
open set.
Any set X with a collection of subsets classified as open
satisfying the three stated properties is called a topological space. For example, the set of real numbers with
unions of open INTERVALs as the open sets is a topological space. So too is the set X = {a,b,c,d} with subsets
Ø, {a}, {b,c,d}, and X as 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; MÖBIUS 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, PAPPUS’S THEOREMS show that the VOLUME V and the surface AREA A
of a torus are given by the formulae:
V = 2π
2
a
2
b
A = 4π
2
ab
where a is the radius of the circle rotated about a line,
and b is 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 CARTESIAN COORDINATE system, if the circle of
radius a is positioned in the yz-plane with center at
position b along the y-axis and is rotated about the zaxis, then the equation of the resulting torus produced
is given by:
This follows from PYTHAGORAS’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 triangle of height z with radius a as its hypotenuse.)
If θ denotes the angle of a point on the generating
circle, and ϕ the 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
x y
b
z
a
2
2
2
2
2
+
−
+
=
500 torus