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In the mansions of mathematics today there is no room for one who
still thinks about the most famous problem of all time - squaring the
circle. For one hundred and ten years mathematicians have been
convinced that the nearly four thousand year old problem of squaring
the circle is sufficiently understood and that it is insoluble. With
a prickliness perhaps betraying some lingering anxiety, anyone who
wastes further time on the puzzle is regarded as mathematically
incompetent.

Quietly demurring from this starkly intimidating judgment stands
ancient Greek civilization itself. They were an extraordinary people,
naming whole branches of knowledge we venerate. Hardly the kind to
waste their time on a fool's errand. One can only marvel at the work
of Apollonius of Perga (247 - 205 BC) on conic sections. What
impelled this great mind to master such an obscure subject that would
have no utility for eighteen hundred years. And then we move forward
and study Isaac Newton's (1642 -1727) Principia Mathematica (1687)
and realize that he could not have made his discoveries about
centripetal forces if he did not have the principles of the ellipse,
parabola and hyperbola with which to build on Johannes Kepler's
(1571-1630) interpretation of Tycho Brae's (1546 -1601) naked eye
measurements of the motions of the planets. And that Kepler himself
first needed Apollonius' conics to derive his laws of planetary
motion from Brae's data. From this rich perspective, the instinct
that prompted the ancient Greek mathematicians to study conics
becomes even more remarkable.

Then when we learn that it was the challenge of squaring the
circle which gave birth to the original interest in conics, something
warns us to be more careful about dismissing as fruitless any matter
the greatest of the Greeks found intellectually important, but
instead strive to reexamine and adjust our own idea of it.

According to historians, leading mathematicians in Greek
antiquity would "occupy" themselves with this geometrical problem,
known as the "quadrature". What it involves, essentially, is
constructing an ideal square with an area equal to that of a given
circle (where the radius of the circle is one, an area equal to pi)
and doing so in a finite number of operations using only a straight
edge and a compass. A practically identical problem is the
rectification of the circle: Constructing an ideal straight line
equal in length to the circumference of the circle.

Beginning with Rene Descartes (1596 - 1650), advances in the
methods of coordinate geometry enabled mathematicians to translate
any geometrical problem into an equivalent algebra problem involving
only numbers and their relations. It was thereby established that a
geometric problem can be solved with a ruler and compass in a finite
number of steps only if its algebraic equivalent depends on a number
that can be obtained from a whole number by addition, subtraction,
multiplication, division or extraction of square roots. There are
numbers that are beyond algebraic, or transcendental; these cannot be
the root of any algebraic or constructible equation. In 1882 it was
proved by Lindemann (1852 -1939) that pi is such a number. Therefore
constructing the long sought for square by means of a finite number
of Euclidean operations alone is impossible. Further pursuit of this
problem - exactly as it has been defined by the historians - is
without question a dead end. But it was a successful failure indeed.
Those who chronicle the "completed" history of the problem recount
the instances where this doomed approach to pi down through the
centuries was nevertheless responsible for important achievements in
the development of mathematics.

Problems, too, can evolve. The problem of squaring the circle has
passed into an ageometric one of understanding the mystery of pi,
seeking some hidden pattern in pi, some design, some relationship
never noticed between the circle and its area the square. This is no
dead end. A man who has been called the most knowledgable
mathematician in 100 years occupied himself with a mail-order
supercomputer calculating pi to 2.26 billion decimal places, looking
for a system. The mathematical intuition that fosters this dedication
must be the same as that which drew the ancient Greek scholars to
this, even then, age old subject. That the Greeks lacked the
essential numeration system of positional decimal notation, let alone
calculators, needed to observe pi in this way argues all the more
eloquently for the importance of instinct in these matters.

And what of their obsolete straight edge and compass, now
discarded in the continuing quest to fathom pi? Someone said there
are no insoluble problems, only misunderstood problems. So it may be
with "squaring the circle". Invariably when the problem is referred
to the term "squaring" the circle is used. Does this perhaps suggest
an intrinsic tension with the notion of a fixed square and a finite
number of steps? Does it invite consideration of a process, of
something dynamic, continuing, animated? Let us suppose the nature of
the problem has indeed been mistaken in a key respect. That of course
pi is a never-ending ratio of the way across a circle to the way
around it, and will not be captured in a fixed square constructed by
the stipulated means. Let us suppose that the true point of the
ancient problem, instead, is to use undivided ruler and compass as
instruments to examine pi by constructing a dynamic square, one that
mirrors the unending decimal expansion of pi. To track pi and express
it in the form of a square with straight edge and compass. We find to
our fascination that there is such a square. It may be said to
vibrate.

Where would one begin to construct a theoretical square
whose area follows along with the area of a forcing circle in the
unending decimal dance of pi? One promising starting point will be
found in the famous golden section of Eudoxus (c. 408-355 BC), which
Kepler ranked as one of the two great treasures of geometry, after
the theorem of Pythagoras. Illustration 1.

If we draw a line from the golden section C to the point D in
Eudoxus' 1:2 right triangle, we may use that line CD as the
semidiameter of an interesting square. Illustration 2. The
area of the square will of course be one-half the square of twice the
semidiameter, or 3.1671845 - not all that bad a starting
approximation of pi, actually. We need a reference point. Imagine a
perfect pi amounting to exactly 3.125. The semi-diameter C'D of a
square of area 3.125 in this construction would start to the right of
the golden section point, at .75 on the base of the 1:2 right
triangle, compared to the section point east of it at .7639320. Let
us call these values "base amounts", referring to the base of the
right triangle CBD that we use to determine the length of the
semidiameter of the sought for square. It will be seen that the
difference between these two points -measuring .0139320 - is the gulf
in which pi reverberates throughout infinitesimal eternity.

Now let this small distance - .0139320 - be the hypotenuse of an
inverted 1:2 right triangle, constructed with straight edge and
compass as in Proposition 12 of Euclid's Elements. Illustration
3. Henceforth, it is possible to follow the successive decimal
expansion of pi, each time marking off a more exact base amount or
starting point for the semidiameter of the pursuing square, merely by
manipulating the 1:2 right triangle form in one of four basic ways.
Through operations constructible with a straight edge and a compass -
and a calculator to give the decimals of pi and confirm the length of
the segments being marked off - one can arrive at ever more accurate
base amount locations for the terminus of the oscillating, note-like
semidiameter.

Only the first two steps need to be explained in order understand
the rest. In the first step, Illustration 4, we draw an
altitude fg to the hypotenuse of the east-facing 1:2 right triangle.
This marks one-fifth of the hypotenuse, gc'. We double that (hc') to
get two-fifths - .0055728 - and this is added to the right hand limit
of .75 to arrive at our third base amount - .7555728. This one step
already gives us pi of 3.1417, more accuracy than Archimedes (287 -
212 BC) achieved using his method of exhaustion involving inscribed
and circumscribed hexagons of 96 sides. The addition here, .0055728,
serves as the hypotenuse for the next 1:2 right triangle in the
series.

It is necessary to reduce the size of the square. To do so, we
face the inverted 1:2 right triangle west, meaning that we are
subtracting from the base amount and the area of the square so
determined. This alternating pattern, east - west, add - subtract,
accounts for the ever diminishing yet unending vibration in the
square. We drop an altitude and mark off a similar 1:2 right triangle
one-fifth the size. Then (moving out from the center line in the
drawing) mark off one-half of that smaller triangle, and one-half
again, and yet a third time. Finally, we swing the altitude of the
last triangle down upon the hypotenuse, recalling the first step in
Eudoxus' original construction. This marks the next base amount,
which gives us pi of 3.141592 - probably more accuracy than any
earthly engineering application would ever require. (The decrement
will form the first hypotenuse in the next step.)

From here on, the appropriate manipulation of the 1:2 right
triangles is shown in the drawing (Illustration 5). In 42
east/west operations (79 steps in all) we have pi to 35 decimal
places. It looks like a series of semaphore signals. (Best to print
Illustration 5.)

The idea of an alternating infinite series of plus and minus
terms approaching pi is not new. The simple series pi/4 = 1/1 - 1/3 +
1/5 - 1/7 + 1/9 . . . is named for Leibniz (1646 - 1716), although it
was known earlier. Histories of this problem have reported no
suggestion by Leibniz or others of such a plus/minus formula for a
dimension of a square (or any other figure) whose area approaches pi,
however. There is also the matter of speed of convergence. As noted,
the second step in the method discussed here (Illustration 6.
gives a formula approaching the square root of [pi/2 - 1] ) yields pi
correct to six decimals.

In the Leibniz series you would have pi to only 3 correct decimal
places after seven thousand alternate steps. The series

[pi - 3]/4 = 1/(2 x 3 x 4) - 1/(4 x 5 x 6) + 1/(6 x 7 x 8)
. . .

gives pi to 6 decimals in about 110 steps. The series

gives 6 places in only 13 steps. But neither do these latter two
series relate to any square. They may be said to play hopscotch back
and forth over pi, but no geometric image is imparted.

Mathematicians have had no success searching for a pattern
throughout more than 2 billion decimal digits of pi. The straight
edge and compass no longer play any role at all in this modern quest.
The kind of method discussed here offers a different vantage point
for observing pi. It reintroduces the classical straight edge and
compass, redefining the problem of squaring the circle to avoid
Lindemann's dead end. Perhaps it will yield something interesting,
maybe even some tool needed for solution of a scientific problem. A
mathematical explanation of why 4 relations of the 1:2 right triangle
can be arranged in an alternating series to approach square root
(pi/2) - 1 would be intrinsically valuable.

Finally, a thought about proportion. If the line AB in
Illustration 2 was the distance light travels in one thousand
years, the increment subtracted in the last step in the series in
Illustration 6 would be near the scale of the subatomic
particle known as the quark. Yet we have barely begun to express pi
with straight edge and compass. The prospect recalls a passage from
the Pensees of Blaise Pascal (1623-1662):

But, to offer him another prodigy equally astounding, let him
look into the tiniest things he knows. Let a mite show him in its
minute body incomparably more minute parts, legs with joints, veins
in its legs, blood in the veins, humours in the blood, drops in the
humours, vapours in the drops; let him divide these things still
further until he has exhausted his powers of imagination, and let the
last thing he comes down to now be the subject of our discourse. He
will perhaps think that this is the ultimate of minuteness in nature.
I want to show him a new abyss. I want to depict to him not only the
visible universe, but all the conceivable immensity of nature
enclosed in this miniature atom. Let him see there an infinity of
universes, each with its firmament, its planets, its earth, in the
same proportions as in the visible world, and on that earth animals,
and finally mites, in which he will find again the same results as in
the first; and finding the same thing yet again in the others, he
will be lost in such wonders, as astounding in their minuteness as
the others in their amplitude. For who will not marvel that our body,
a moment ago imperceptible in a universe, itself imperceptible in the
bosom of the whole, should now be a colossus, a world, or rather a
whole, compared to the nothingness beyond our reach? Anyone who
considers himself in this way will be terrified at himself, and,
seeing his mass, as given him by nature, supporting him between these
two abysses of infinity and nothingness, will tremble at these
marvels. I believe that with his curiosity changing into wonder he
will be more disposed to contemplate them in silence than investigate
them with presumption.