If 865 points are scattered on a square sheet of
paper 1 ft wide, then some seven of those
points will be clustered sufficiently close so as
to lie in a square 1 in. wide.
(Divide the sheet of paper into 144 squares each of
side-length 1 in. and label each point by the square in
which it lies. For points that fall on a line between two
squares, arbitrarily decide to which square it belongs.
As 865 = 144 ×6 + 1, one of these small squares con-
tains seven points.)
The following principle is sometimes considered a
variation of the pigeonhole principle:
If nnumbers sum to S, then not all the numbers
are larger than . Also, not all the numbers are
smaller than .
As a simple consequence we have:
If wages of five workers summed to $100,000,
then at least one worker earned no more than
$20,000.
See also
COMBINATORICS
.
place-value system See
BASE OF A NUMBER SYSTEM
.
plane Informally, a plane is a flat surface having no
edges and extending infinitely in all directions. More
precisely, a plane is a geometric surface, without edges,
with the property that any straight line connecting two
points in the surface remains in the surface.
Planes are regarded as two-dimensional—within
the surface there are only two independent directions
of motion: “left and right” and “back and forth.” Any
other motion (diagonal motion, for example) can be
thought of as a combined effect of these two motions.
Sitting in three-dimensional space, a plane is
thought of as a geometric object with no thickness. It is
possible to write down an equation for such a plane.
(See
VECTOR EQUATION OF A PLANE
.)
Plane geometry is the study of relationships between
points, lines, and curves lying in the same plane. A plane
figure is a figure, such as a square, circle, or a triangle,
that lies in a plane. Points that all lie on the same plane
are called coplanar.
Two planes sitting in three-dimensional space inter-
sect if they share points in common. If they do, the set
of points they share forms a straight line. Two planes
that do not intersect are called parallel. Three planes
sitting in three-dimensional space can intersect in a
point, along a line, or not at all.
In both two- and three-dimensional space, two
points Pand Qdetermine a line. In three-dimensional
space, three points P, Q, and Rdetermine a plane
(under the proviso that P, Q, and Rdo not themselves
lie on a straight line). To see this, consider the line that
passes through Pand Q. There are infinitely many
planes that contain this line. The point Rdetermines
which of these planes to choose. (Alternatively, using
VECTOR
analysis, the
POSITION VECTOR
s→
PR and
→
QR are
two vectors in the desired plane, and so their
CROSS
PRODUCT
n=→
PR ×
→
QR is a normal to the plane. This is
all we need to write down the
VECTOR EQUATION OF A
PLANE
.) Any three points in three-dimensional space
are coplanar.
Four points might or might not all lie on the same
plane. This is the reason why four-legged chairs are
unlikely to be stable on rough, uneven floors. The tips
of the chair legs represent four coplanar points, but the
locations at which you wish to place these points on an
uneven floor might not be. A three-legged stool, how-
ever, will always be stable, as any three points on the
uneven ground will be coplanar. For this reason, barn
stools used for milking, say, are traditionally made
three-legged.
Plato (ca. 428–348
B
.
C
.
E
.) Greek Philosophy Born
in Athens, Greece, philosopher Plato is remembered in
mathematics for promoting the notion that mathemati-
cal concepts have a real existence independent of
human thought. This is consequently linked to the
argument that mathematicians discover mathematics,
rather than create it. Plato deemed mathematics as an
essential part of a valued education, and he greatly
influenced the esteem with which mathematics is
regarded in the Western world. He was also the first to
describe the five P
LATONIC SOLID
s.
Plato was a worldly man very much interested in
politics and human affairs. He was a member of the
military service during the Peloponnesian War
S
–
n
S
–
n
Plato 395