was the task of developing a general procedure that
would divide any given angle into three equal parts. The
origin of this problem is unknown, though it seems a natural question to ask given that the procedure of bisecting
any angle is relatively straightforward. (Given an angle
defined by two rays, draw the arc of a circle centered at
the vertex of the angle to find two points on the rays at
an equal distance from this vertex. Now draw circles of
fixed radius with centers at each of these two points. The
line connecting the vertex of the angle to the point of
intersection of the two circles is an angle bisector.)
Using a compass and a ruler, that is, a straightedge
with specific lengths marked along it, HIPPOCRATES OF
CHIOS (ca. 440 B.C.E.) developed a straightforward
general solution to the problem, as did ARCHIMEDES
OF SYRACUSE (ca. 287–212 B.C.E.) some 200 years
later. Around this time, for the sake of increased intellectual challenge, scholars decided to add the restriction that only the primitive tools of a straightedge with
no markings and a compass could be used in the solution. The difficulty of the problem increased significantly. APOLLONIUS OF PERGA (ca. 262–190 B.C.E.)
found a solution to the problem using only the basic
tools under the assumption that a HYPERBOLA could be
drawn. Unfortunately, this was not deemed an admissible assumption. The problem remained one of the
greatest unsolved challenges for two millennia.
In the early 1800s mathematicians focused on the
specific problem of trisecting an angle of 60°, or, equivalently, the problem of constructing an angle of 20°
using straightedge and compass alone. In particular, if
such an angle can be produced, then it is possible to
construct a line segment of length x = cos20 (as one of
the legs of a right triangle with angle 20°). The trigonometric identity cos(3θ) = 4cos
3
(θ) – 3cos(θ), noting that
cos(60) = 1/2, shows that the length x must satisfy the
equation: 8x
3 – 6x – 1 = 0.
The theory of CONSTRUCTIBLE numbers shows that
any quantity of rational length can be constructed with
straightedge and compass, and that if two lengths l 1
and l 2 can be produced, then so too can their sum, difference, product, and quotient, along with the square
root of each quantity. Mathematicians showed that any
solution x to the equation above cannot be rational
and, moreover, in 1837, French mathematician Pierre
Laurent Wantzel (1814–48) proved any such number x
is not constructible. Thus the general problem of trisecting an angle is unsolvable.
It is interesting to note that in the field of origami,
using only the tool of paper folding, it is possible to trisect any given acute angle. Assume the angle is placed in
the bottom left corner A of a square piece of paper, with
one ray defining the angle being the bottom edge of the
sheet, and the other ray, call it L, a crease in the paper.
Fold a crease of some arbitrary height parallel to the
bottom edge and fold a second parallel crease half this
height. Call the distance between these two creases a,
and the point on the left edge at height 2a point B. Take
the bottom left corner of the sheet and fold the paper so
that point B lies on the crease L (call the location of this
point C) and so that the point A lies on the crease at
height a, to define a point D. The line connecting A to
D is precisely one-third of the original angle.
(To see why this works, notice that PYTHAGORAS’S
THEOREM establishes that line segments AD and BD
have the same length. By the symmetry of the folding,
length AC is the same as length AD, establishing that
triangle CAD is isosceles. Draw the bisector of this triangle from A. This produces a diagram with three congruent right triangles, showing that the angle at A is
indeed divided into three equal parts.)
Tristram Shandy paradox This paradox about the
infinite is derived from Lawrence Sterne’s 1760 novel
Tristram Shandy, which purports to be part of the protagonist’s autobiography. In it, the hero Shandy observes
that it has taken him two years to describe his first two
days, and so concludes that it will be impossible for him
to ever complete the autobiography in full. Philosopher
and mathematician BERTRAND ARTHUR WILLIAM RUSSELL (1872–1970) pointed out that if the author were
immortal, however, he would be able to complete his
goal, still writing at the same rate of progress. (Of
course, it would take an infinite amount of time to do
so.) Russell argued that a life of infinite length contains
just as many years as it does days.
See also CARDINALITY; HILBERT’S INFINITE HOTEL;
INFINITY; PARADOX.
trivial solution Any solution to a problem that is
obvious, or of no interest in the given context, is called
a trivial solution. For example, the famous equation of
FERMAT’S LAST THEOREM x
n + y
n = z
n has trivial solutions x = 0, y = 0, z = 0 and x = 1, y = 0, z = 1, and the
trivial solution 513
would divide any given angle into three equal parts. The
origin of this problem is unknown, though it seems a natural question to ask given that the procedure of bisecting
any angle is relatively straightforward. (Given an angle
defined by two rays, draw the arc of a circle centered at
the vertex of the angle to find two points on the rays at
an equal distance from this vertex. Now draw circles of
fixed radius with centers at each of these two points. The
line connecting the vertex of the angle to the point of
intersection of the two circles is an angle bisector.)
Using a compass and a ruler, that is, a straightedge
with specific lengths marked along it, HIPPOCRATES OF
CHIOS (ca. 440 B.C.E.) developed a straightforward
general solution to the problem, as did ARCHIMEDES
OF SYRACUSE (ca. 287–212 B.C.E.) some 200 years
later. Around this time, for the sake of increased intellectual challenge, scholars decided to add the restriction that only the primitive tools of a straightedge with
no markings and a compass could be used in the solution. The difficulty of the problem increased significantly. APOLLONIUS OF PERGA (ca. 262–190 B.C.E.)
found a solution to the problem using only the basic
tools under the assumption that a HYPERBOLA could be
drawn. Unfortunately, this was not deemed an admissible assumption. The problem remained one of the
greatest unsolved challenges for two millennia.
In the early 1800s mathematicians focused on the
specific problem of trisecting an angle of 60°, or, equivalently, the problem of constructing an angle of 20°
using straightedge and compass alone. In particular, if
such an angle can be produced, then it is possible to
construct a line segment of length x = cos20 (as one of
the legs of a right triangle with angle 20°). The trigonometric identity cos(3θ) = 4cos
3
(θ) – 3cos(θ), noting that
cos(60) = 1/2, shows that the length x must satisfy the
equation: 8x
3 – 6x – 1 = 0.
The theory of CONSTRUCTIBLE numbers shows that
any quantity of rational length can be constructed with
straightedge and compass, and that if two lengths l 1
and l 2 can be produced, then so too can their sum, difference, product, and quotient, along with the square
root of each quantity. Mathematicians showed that any
solution x to the equation above cannot be rational
and, moreover, in 1837, French mathematician Pierre
Laurent Wantzel (1814–48) proved any such number x
is not constructible. Thus the general problem of trisecting an angle is unsolvable.
It is interesting to note that in the field of origami,
using only the tool of paper folding, it is possible to trisect any given acute angle. Assume the angle is placed in
the bottom left corner A of a square piece of paper, with
one ray defining the angle being the bottom edge of the
sheet, and the other ray, call it L, a crease in the paper.
Fold a crease of some arbitrary height parallel to the
bottom edge and fold a second parallel crease half this
height. Call the distance between these two creases a,
and the point on the left edge at height 2a point B. Take
the bottom left corner of the sheet and fold the paper so
that point B lies on the crease L (call the location of this
point C) and so that the point A lies on the crease at
height a, to define a point D. The line connecting A to
D is precisely one-third of the original angle.
(To see why this works, notice that PYTHAGORAS’S
THEOREM establishes that line segments AD and BD
have the same length. By the symmetry of the folding,
length AC is the same as length AD, establishing that
triangle CAD is isosceles. Draw the bisector of this triangle from A. This produces a diagram with three congruent right triangles, showing that the angle at A is
indeed divided into three equal parts.)
Tristram Shandy paradox This paradox about the
infinite is derived from Lawrence Sterne’s 1760 novel
Tristram Shandy, which purports to be part of the protagonist’s autobiography. In it, the hero Shandy observes
that it has taken him two years to describe his first two
days, and so concludes that it will be impossible for him
to ever complete the autobiography in full. Philosopher
and mathematician BERTRAND ARTHUR WILLIAM RUSSELL (1872–1970) pointed out that if the author were
immortal, however, he would be able to complete his
goal, still writing at the same rate of progress. (Of
course, it would take an infinite amount of time to do
so.) Russell argued that a life of infinite length contains
just as many years as it does days.
See also CARDINALITY; HILBERT’S INFINITE HOTEL;
INFINITY; PARADOX.
trivial solution Any solution to a problem that is
obvious, or of no interest in the given context, is called
a trivial solution. For example, the famous equation of
FERMAT’S LAST THEOREM x
n + y
n = z
n has trivial solutions x = 0, y = 0, z = 0 and x = 1, y = 0, z = 1, and the
trivial solution 513