figures. (If, for instance, three points A, B, and Crepre-
sent the vertices of an equilateral triangle of side-length
1, then the images A′, B′, and C′of the three points
under an isometry again form an equilateral triangle of
side-length 1.)
The
FUNDAMENTAL THEOREM OF ISOMETRIES
shows that any isometry in the plane is the result of at
most three reflections. Any isometry that preserves the
location of three planar points leaves all points in the
plane unchanged. No isometry can preserve the loca-
tion of only two points. If an isometry preserves the
location of just one point, then that isometry must be a
rotation about that point.
isomorphism A one-to-one and onto correspondence
between the elements of two sets equipped with
BINARY
OPERATION
s that preserved the operations on those sets
is called an isomorphism. For example, the map f:n →
2nmatches each natural number with an even number
(and, in reverse, each even number is paired with a
unique natural number) and, moreover, preserves the
operation of addition between the two sets: We have f
(n+ m) = f(n) + f(m) for all natural numbers nand m.
Thus, with respect to addition, fis an isomorphism
between the two sets. (Note, however, that fdoes not
preserve products and so is not an isomorphism with
respect to multiplication.)
The
LOGARITHM
function ln xprovides a map
between the set of all positive real numbers and the set of
all real numbers. The logarithm converts multiplications
in the first set into additions in the second set: ln(xy) = ln
x+ ln y. The set of positive reals under multiplication is
thus isomorphic to the set of all reals under addition.
The word isomorphism is derived from the Greek
words isos meaning “same” or “equal” and morphos
meaning “shape” or “structure.”
See also
HOMOMORPHISM
.
isoperimetric problem A classical problem in math-
ematics, called the isoperimetric problem, asks:
Of all figures drawn in the plane with a given
perimeter, which encloses the largest area?
The origin of this problem dates back to antiquity
with the famous legend of Princess Dido and D
IDO
’
S
PROBLEM
.
The topic of isoperimetrics (prefix iso: the same)
was systematically studied by the 17th- and 18th-cen-
tury Swiss family of mathematicians, the B
ERNOULLI
FAMILY
. They discovered and classified many curves
satisfying certain maximum and minimum properties,
but did not solve the isoperimetric problem. L
EON
-
HARD
E
ULER
(1707–83) also contributed to this field,
using the techniques of
CALCULUS
.
J
AKOB
S
TEINER
(1796–1863) used purely geometric
methods to establish the following partial answer: if
there were a shape that offered maximal area for a
given perimeter, then that shape must be a circle.
Steiner thought that he had completely solved the
isoperimetric problem, that the answer must be a circle,
but several years later German mathematician K
ARL
T
HEODOR
W
ILHELM
W
EIERSTRASS
(1815–97) pointed
out that the assumption that problems always have
solutions can lead to absurdities. The following amus-
ing example illustrates his concern:
Consider the question: of all positive integers,
which is the largest? If there is an answer to
this problem, then it cannot be a number ndif-
ferent from 1, for then n2is an integer larger
than n. This leaves only the integer 1 as the
answer to the question.
Weierstrass realized that it could very well be the case
that the isoperimetric problem has no solution, despite
Steiner’s work, and so perhaps even the circle can be
replaced with a shape of greater area for the same
perimeter. To completely solve the isoperimetric prob-
lem, Weierstrass realized one must also prove that a
solution to the problem exists. Finally, in 1870, Weier-
strass was able to do this by developing a new mathe-
matical theory today called the “calculus of variations.”
The solution to the problem, known by ancient schol-
ars, was, at long last, established as mathematically
correct.
Simple variations of the isoperimetric problem
are easier to handle. Consider, for example, the ques-
tion: of all rectangles of perimeter 40 ft, which has
the largest area? This can be answered via an exercise
in algebra: Any rectangle has two long sides and two
short sides. If the perimeter is 40 ft, then two sides
are longer than 10 ft, and two are shorter than 10 ft
(unless the figure is a square). Writing the dimensions
of the rectangle as 10 + xft long, and 10 – xft wide,
for some number x, its area is then given by the formula
284 isomorphism