simple idea that, although any loop drawn on the sur-
face of a
SPHERE
can be continuously shrunk to a point
(while remaining on the surface of the sphere), the
same is not true for loops drawn on the surfaces of
other three-dimensional objects. For example, not all
loops drawn on the surface of a
TORUS
are homotopi-
cally equivalent to a point. Thus the study of loops on
surfaces leads to a method for distinguishing surfaces.
(This is important, for instance, if one is trying to
determine the shape of mathematical objects that can-
not be visualized.) Poincaré conjectured that any sur-
face that has the same homotopy properties of a sphere
is topologically the same as a sphere. This result,
known as Poincaré’s conjecture, has long been known
to hold for one- and two-dimensional surfaces, and,
since 1982, for four- and higher-dimensional surfaces.
Work on proving the conjecture true for three-dimen-
sional surfaces continues today.
From 1892 to 1899 Poincaré published his three-
volume work Les méthodes nouvelles de la méchanique
céleste (New methods in celestial mechanics), a land-
mark piece that aimed to characterize all possible
motions of mechanical systems.
He was appointed chair of the faculty of science at
the University of Paris in 1881, chair of mathematical
physics and probability at the University of Sorbonne
in 1886, and, in the same year, chair at the prestigious
École Polytechnique. He held these chairs until his
death in 1912. In 1887 Poincaré was elected, sepa-
rately, to all five sections of the Académie des Sciences
(mechanics, physics, geography, navigation, and geom-
etry), and in 1908 he was also elected to the Académie
Française. He also received an extraordinarily large
number of awards and honors from other learned insti-
tutions around the globe, including the R
OYAL
S
OCIETY
S
YLVESTER
M
EDAL
in 1901 and the Bruce Medal of the
Astronomical Society of the Pacific in 1911.
Poincaré died in Paris, France, on July 17, 1912.
point A location in space (or on a curve or a surface)
is called a point. A point is usually specified by its
COORDINATES
in a coordinate system. For instance, a
point in a three-dimensional system of C
ARTESIAN
COORDINATES
is given by a triple of numbers (x,y,z).In
elementary
GEOMETRY
, a point is an undefined term
but is loosely thought of as a geometric entity having
no dimensions. The geometer E
UCLID
(ca. 300
B
.
C
.
E
.)
gave the vague description of a point as “that which
has no breadth.”
A collection of lines is said to be
CONCURRENT
if
the lines meet at a common point. A
TRIANGLE
pos-
sesses many interesting points of concurrency.
In arithmetic, the decimal point is the symbol used
to separate the integer part of a number from its frac-
tional part. For example, the number 2.53 is read as
“two point five three.”
See also
DECIMAL REPRESENTATION
;
NINE
-
POINT
CIRCLE
;
PLOT
.
point of contact (tangency point) The single point
at which two curves, or two curved surfaces, touch, but
do not cross, is called a point of contact. For example,
the point at which a ball sitting on a table touches the
surface of the table is the point of contact of a
SPHERE
and a
PLANE
.
See also
TANGENT
.
Poisson, Siméon-Denis (1781–1840) French Proba-
bility theory Born on June 21, 1781, in Pithiviers,
France, scholar Siméon-Denis Poisson is remembered
for his fundamental work on the theory of
PROBABILITY
,
for the discovery of the distribution named after him,
and also for his formulation of the
LAW OF LARGE NUM
-
BERS
, all completed in the latter part of his mathemati-
cal career. Before then Poisson had made significant
contributions to the topic of celestial mechanics and to
the theory of electricity and magnetism.
Poisson entered the École Polytechnique in Paris in
1798 to study mathematics under the guidance of
J
OSEPH
-L
OUIS
L
AGRANGE
(1736–1813) and P
IERRE
-
S
IMON
L
APLACE
(1749–1827), both members of the
faculty at the time. After graduating just two years
later, Poisson was granted a deputy professorship at the
institution and later a full professorship in 1806.
During this early period of his career Poisson stud-
ied
DIFFERENTIAL EQUATION
s as well as
POWER SERIES
and their applications to mechanics and physics. Start-
ing in 1808 he produced fundamental results extending
the work of Laplace and Lagrange on the motion of the
planets, developing new series techniques to approxi-
mate solutions to perturbations in their orbits. He pub-
lished an influential two-volume treatise on the topic of
mechanics in 1811 and solved important problems on
Poisson, Siméon-Denis 399