parabolas is called the focus of the paraboloid. The
reflection property of a parabola shows that light
rays and beams of sound emitted from the focus of a
parabola are reflected out of the paraboloid as a par-
allel beam. Reflectors in automobile headlights, for
example, are parabolic. Conversely, a parallel beam
received by a paraboloid reflector is converged into
its focus. Satellite dishes and reflecting telescopes use
parabolic reflectors.
An elliptic paraboloid has horizontal cross-sec-
tions that are
ELLIPSE
s and is given by the equation z=
ax2+ by2, where aand bare constants. A hyperbolic
paraboloid (also called a saddle surface) satisfies z=
ax2– by2. It has horizontal cross-sections that are
HYPERBOLA
s, and indeed resembles the shape of a
horse saddle.
See also
ELLIPSOID
;
HYPERBOLOID
.
paradox A statement that seems contradictory or
counter to common sense is called a paradox. Some
paradoxes in mathematics arise as logical conse-
quences of seemingly plausible premises, such as the
Banach-Tarski paradox arising from the study of
AREA
, B
ERTRAND
’
S PARADOX
, and R
USSELL
’
S PARADOX
.
(The appearance of these particular paradoxes caused
mathematicians to reassess the basic foundations of
mathematics.) Other paradoxes can be simple self-con-
tradictory statements. For example, it is said that
Socrates once reported, “One thing I know is that I
know nothing,” and George Orwell once wrote, “Free-
dom is slavery.” Both assertions have a flavor similar
to the
LIAR
’
S PARADOX
.
Other famous paradoxes include:
Grelling’s Paradox
Some words in the English language aptly describe
themselves. For example, pentasyllabic, meaning “hav-
ing five syllables”, is indeed pentasyllabic. Other words
are not self-descriptive: for example, monosyllabic, edi-
ble, and tangible. Define a new word heterological to
mean “a word that does not describe itself.” Now ask,
“Is heterological heterological?”
Berry’s Paradox
The number one million, one hundred thousand, one
hundred and twenty one can be named as “the first
number not nameable in under 10 words.” (All num-
bers less than 1,100,121 can be named in nine words
or less.) But this latter description uses only nine words
to indicate this number and so it can be named with
less than 10 words, despite its definition.
The Barber Paradox
A barber places a sign on the store window. It reads:
“This barber shaves all men who do not shave them-
selves, and only these men.” Who shaves the barber?
The Lawyer Paradox
This paradox is traditionally ascribed to the Greek
philosopher Protagoras (490–421
B
.
C
.
E
.)
A law instructor accepts a penniless student under
his wing for tuition under the agreement that the stu-
dent pay the tutor his fees if and only if he wins his
first case in court. However, after qualifying as a
lawyer, the student takes up a different career and
never undertakes a first case. The tutor later sues him
for his fees.
The student cleverly decides to represent his own
case. This way, he reasons, he need never pay the tutor
his fees. If he wins the case, the ruling shall be that he
need not pay, whereas if he loses the case, he would be
exempted from paying as per his previous agreement
with the tutor. Surprisingly, the tutor reasons too that
he cannot lose. If the student wins this case, then he
must pay the fees according to their previous agree-
ment, whereas, if the student loses, the ruling shall be
that he must pay!
The Unexpected Quiz
A school-teacher informs his class that there shall be a
surprise quiz some day the following week, and that no
one shall be able to deduce the night before that the quiz
shall come the next day. Students then reason as follows:
There can be no quiz on Friday, for on Thurs-
day night, having had no quiz the four previ-
ous days, we would know the quiz will come
next day. The quiz must therefore be some day
other than Friday.
The quiz cannot occur on Thursday either,
for on Wednesday night, having had no quiz
the three previous days (and with Friday ruled
out as a possibility) we would know that the
quiz will come next day.
By the same reasoning, the quiz cannot
occur on Wednesday, Tuesday, or even Monday.
374 paradox