212 fundamental theorem of isometries

note that is an antiderivative of the

function in question, F′(x) = x2. The desired area is

thus units squared. The entire

thrust of

INTEGRAL CALCULUS

is consequently trans-

formed to the study of

DIFFERENTIAL CALCULUS

employed in reverse.

See also

ANTIDIFFERENTIATION

.

fundamental theorem of isometries Let ABC be a

triangle in the plane. Then the location of any point P

in the plane is completely determined by the three num-

bers that represent its distances from the vertices of the

triangle. Any

GEOMETRIC TRANSFORMATION

that pre-

serves distances is called an isometry. If an isometry

takes points A, B, and Cto locations A′, B′, and C′,

respectively, then it also takes the point P to the unique

point P′with matching distances from A′, B′, and C′.

Thus an isometry is completely determined by its effect

on its vertices of any triangle in the plane.

Three

REFLECTION

s, at most, are ever needed to

map three vertices A,B, and Cof one triangle to three

vertices A′, B′and C′of another congruent triangle.

(First reflect along the perpendicular bisector of AA′to

take the point Ato the location A′. Next, reflect along

a line through A′to take the image of Bto B′. One

more reflection may be needed to then send the image

of Cto C′.) This observation proves the fundamental

theorem of isometries:

Every isometry of the plane is the composition

of at most three reflections.

fuzzy logic In 1965, Iranian electrical engineer Lofti

Zadeh proposed a system of logic in which statements

can be assigned degrees of truth. For example, whether

or not Betty is tall is not simply true or false, but more

a matter of degree.

Fuzzy-set theory assigns degrees of membership to

elements of fuzzy sets. These degrees range from 1,

when the element is in the set, to zero when it is out of

the set. Betty’s membership in the set of tall people is a

matter of degree.

See also

LAWS OF THOUGHT

.