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Squaring the
Circle
by
Miles Mathis
painting
by Jake Baddeley
It is commonly claimed
that it has been proved, first by Lindemann, that the circle
cannot be squared. I will show that this is false.
What has been proved by Lindemann et. al is that π is
transcendental and therefore cannot be measured with absolute
accuracy by a compass. This much is true and I therefore have
nothing to say against it. Since a transcendental number is
expressed by an infinite series, any given length achieved by a
compass will only be an approximation of that number. However, I
would point out that no
real physical length, rational or irrational, algebraic or
transcendental, may be drawn with absolute accuracy. Every
possible length, even the natural number 1, must be an
approximation when drawn with real instruments. This is due to
the very simple fact that real instruments have size.
The point of the compass or pencil, or even the width of the
laser, must have some size. In real measurement, we cannot
measure at zero; we cannot find a mathematic point. There will
always be a necessary logical separation between a mathematical
point and a real point.
What this means for squaring the
circle is that in the real world it is quite possible to square
the circle using a compass. Using a standard metal compass, I can
square the circle to the precision of my instrument in a very
short time, using only the first few terms in my infinite series
expression of π. Even if my compass point is very sharp, I only
need a handful of my infinite terms before I have exhausted my
available precision. Meaning that I will have found the length of
π to an accuracy that is smaller than the width of the point.
Even if you give me a laser compass I will be able to provide you
with an approximation that exceeds the accuracy of the
instrument, and I can do this in real time. That is to say, I can
provide you with a real physical length for π that is correct to
within an atom, and I won’t even need a computer to do it. I
can do it just adding lengths with my laser compass.
Let us use the Basel problem to show this: ζ(2) = ^{∞}Σ_{n=1}(1/n^{2})
= 1/1^{2} + 1/2^{2}
+ 1/3^{2} +
1/4^{2}... = π^{2}/6
If our dimension is the centimeter, then how many terms do we
need to express π to within the width of an atom? If an atom has
a diameter of 10^{8}cm,
then I will need on the order of ten thousand terms to achieve
that accuracy. If I can measure off ten lengths per minute with
my laser compass and microscope, then it will take me 1000
minutes to square the circle, or almost 17 hours. Quite a chore I
grant you, but far from impossible.
Only if we require accuracy down to the Planck length does
squaring the circle look impossible, but even then it is only a
matter of time and speed. The Planck length is about 10^{33}cm,
which requires us to take the equation out to the 10^{16}th
term of the series. This would take me 2 billion years to
accomplish at ten per minute. But such a task would be a simple
one for a computer, which can accomplish 36 trillion calculations
per second. In a computer, a calculation is
a physical task. This means that a task like measurement could
also be speeded up drastically, making π measurable to within a
Planck length in real time. All this is almost beside the point,
however, since the original question of squaring the circle was
posed by the Greeks, who had no conception of the Planck length.
They had a conception of the atom but no idea of requiring
accuracy of measurement in geometry down to the atomic width.
The fallacy of the squaring of the circle has been in
thinking of natural or rational numbers as definite, and thinking
of irrational or transcendental numbers as indefinite. What I
mean by this is that it is falsely assumed that a line on a piece
of paper with the length 1 has a definite and precise length
where a line with a length of π does not. The truth is that they
both have definite
and precise lengths mathematically, and they both
have indefinite and imprecise lengths physically or in real
measurement. As proof of the
first part of this assertion, consider the circumference of the
circle. There is no imprecision in that length, mathematically.
If you walk that curve, there must be a definite point when you
have returned to the place you started. That point can be
measured with any accuracy you desire. That is, it has a
potentially infinite precision. However accurate you want to be,
you can be. You can shrink yourself down as small as you like,
and you will never get too small to achieve a significant number.
Just like the number one, the length of that circumference is
absolutely and infinitely exact. The problem, therefore, is not a
mathematical one, it is a physical one. Because
the length is infinitely precise, it cannot be measured with
infinite precision in real time.
And this brings us to the second part of my assertion. It is
commonly and intuitively felt that the number 1 is precise and
exact and definite. It holds a place on the number line that is
not fuzzy in the least. But physically
it is just as fuzzy as π or any other transcendental number. As
proof of this, consider being asked to verify that a given line
is in fact precisely 1 meter in length. There are two ways to do
this. One is to compare it to a standard meter bar. The other is
to define the meter as x number of some given quanta, which
quantum is currently defined by the second and the speed of
light. In either case you must fix one end of your given length
and study the other end at high magnification. This means that
although the length 1 may mathematically be very precise, your
measurement of it cannot exceed your magnification. Just as with
the length π, your precision is determined by your instruments,
not by whether the length is rational, irrational or
transcendental. In fact, there
is no geometric operational
difference between rational, irrational and transcendental
numbers. Mathematically they are all absolutely precise, with no
fuzziness in the least. Transcendental numbers occupy
mathematical
points on the number line that are exactly as determinate as
points occupied by rational numbers and natural numbers. The
length of a line π meters long is no more or less fuzzy than the
length of a line 1 meter long. But physically, all numbers of all
sorts are limited in accuracy by the operation of measurement.
And they are limited to the same extent. The number 1 measured to
an accuracy of ± .0001cm is no different than the number π
measured to that accuracy. If you claim that because π is
transcendental, the circle cannot be squared, I can respond that
although the number 1 is natural and rational, and although your
circle has a radius of 1, you cannot copy that circle. You cannot
precisely copy that circle for the same reason I cannot square it
precisely, the reason being that neither one of us can make real
measurements to zerowidth points—or measure to an infinite
precision. I cannot do it due to time considerations, and you
cannot do it due to length considerations (magnification
considerations) but neither of us can do it.
This being true, it is inconsistent to claim that circles cannot
be squared. Unless you want to go on to claim that no
measurements can be made and that nothing can truly be done, then
your argument is a nonstarter. The circle can be squared as
accurately as you can do anything that you admit can be done,
therefore claiming it cannot be done is just foolish.
As one final proof of this, remember that Einstein showed us the
equivalence of time and length, x and t. According to current
theory, this is because time and length are mathematically and
physically transformable. In General Relativity, time is a vector
that is symmetrical to the three length vectors, so that they all
may be expressed by the same sort of variable or function in a
fourvector Gaussian field. In my theory this idea is
considerably simplified by noticing that time, as a measured
quantity, is always a second measurement of x, y or z.
Operationally, time is always reducible to a predefined distance
measurement, which makes it a wholly dependent variable. What
this means for my argument above is that my final statement—that
precisely squaring the circle is impossible for reasons of time
and precisely copying the circle is impossible for reasons of
magnification—has an even greater impact. Time considerations
are length
considerations, and vice versa. For example, the reason my
computer is ultimately limited in the number of calculations or
tasks it can accomplish per minute is that each task is composed
of some tiny motion, and this motion
is limited by the speed of light, and the speed of light is
caused by some distance
the light must travel. Every time limitation is ultimately a
distance limitation. And this means that there is no
difference between the reason I cannot precisely square the
circle and the reason you cannot precisely copy it. Which means
that it must be completely baseless, on logical grounds, to claim
that the circle cannot be squared.
I have shown: 1) If
I can draw a line 1 unit long to a given accuracy, then I can
draw a line π units long to that same accuracy, given enough
time or speed. 2) If I cannot draw a line π units long to a
given accuracy, then I cannot draw a line 1 unit long to that
accuracy. If I cannot draw π to the given accuracy, it is
because I cannot see to that accuracy. If I cannot see to that
accuracy, I cannot draw any line to that accuracy, even a unit
line. But in the historical problem, it is given that I can draw
a line 1 unit long. I am given
a circle with unit radius. This must
mean that I can draw the unit 1 with some given precision, but
cannot mean that I
can draw the unit 1 with infinite precision, since it is
impossible to mark off that unit with infinite precision with any
given compass. I must mark off that unit with some real compass
in order to proceed to compose the length π, therefore I can
assume that my precision is limited in both instances—and
further that it has the same limit.
Therefore I must be able to
square the circle, and the precision of my square (and the time
it takes to draw it) will depend not on the transcendental nature
of π but on the kind of compass you provide. The only thing I
cannot do in finite time is square the circle perfectly, to
infinite precision. But I cannot draw any
geometric figure to infinite precision in finite time, so that my
disability with squaring the circle is not to the point.
It
may be argued against me that copying a circle requires far fewer
tasks than squaring a circle. Every task introduces a margin of
error or imprecision. To copy a circle with a compass requires me
to fix the given radius. This fixing is accurate only to the
width of my point. Then I must transfer that length to a second
location, which introduces a second margin of error of the same
magnitude. We add these two errors for a total potential error
that is twice the width of my compass point. But squaring the
circle requires hundreds, or maybe millions of tasks, creating a
total potential error that may exceed the length of the line
being constructed. This is
true as far as it goes, but it requires further analysis. To
achieve this total potential error one must assume that you, the
constructor, cannot tell any difference between the point of your
compass and a zerodimension point. We must be given two things:
1) that the compass had some width, 2) that you cannot
distinguish that width at all. If we are not given both these
things, then it is possible to assume that you can, by looking
more closely, correct for a large part of that total potential
error. One might assume that you can see that the point hole made
by your compass has a top and a bottom, and by aligning not just
the holes, but the top edges of the holes, you may dispense with
most of your error. You can
already see that it is once again a matter of magnification. It
is a matter of your tools, and one of those tools is your eyes.
How sharp is that edge of the hole? How accurate can you be? The
question is not decided by the compass alone, and is certainly
not decided by the rationality or irrationality of your number.
It is decided by a combination of eye and tool. If you give me
strong enough eyes or a powerful enough microscope, then each
task will have margin of error that is always kept below the
width of the compass. Even the sum of millions of those tasks can
be kept below the width of the compass. This being the case, it
once again becomes foolish to claim that the circle cannot be
squared. I can square a circle to any precision you demand, and
if I can’t it will be due to the weakness of my eyes, not to
the transcendental nature of π.
One final point: some
will say that squaring the circle was not originally a problem of
real construction with a real compass. It was a problem of
abstract geometry, where all lines and points are considered to
be perfect abstractions—where each line had no width and each
point had no extension and a compass was a perfect compass, used
in the mind only and creating no possible margin of error. In
this case, π was bothersome because it could not be so easily
abstracted. The length π retained a fuzziness until you took it
to infinity. This made it a different sort of abstraction than
the natural numbers, which stood for distinct lengths without the
need to exhaust any infinite series.
I actually agree with this point, as an historical point,
although I have two things to say against it. 1) If it is true,
then why have all current explanations rushed to add the proviso
that they mean “constructible lines” and that they are
talking about squaring the circle with a straight edge and
compass? I have just shown that making it a physical problem
assures that it is soluble to any given precision, so that making
it physical was a mistake. It should have been kept abstract,
using an abstract perfect compass. 2) Even if we use an abstract
perfect compass, it is difficult to understand the unease this
problem caused the Greeks. If we are imagining things here—and
that is what abstraction is—why not imagine the infinite series
exhausted? Is it really any harder to
imagine π’s infinite series exhausted than it is to imagine a
line without width or a point without extension? A point without
extension is another infinite series exhausted. It is
the infinite regression toward zero exhausted. The same can be
said for a line or any other abstract
geometric figure. A circle is the infinite progression of perfect
polygons exhausted. The Greek had no problem postulating the
circle in this problem, or the line that is the radius, or the
point that is the end of the radius, but he balked at the length
π. There is a gross inconsistency in
logic here from the very beginning. By the same logic that you
construct a perfect line in your head you can construct the
length π. To imagine an abstract perfect line, you first imagine
a real line, then you make it thinner, then thinner. Then you
exhaust thinness. Constructing π in your head is no different.
You create some routine with your compass and then exhaust that
routine.* It is highly inconvenient that you can’t evenly
divide π and construct it in some direct manner, but this
inconvenience is hardly to the point. It is inconvenient that you
can’t divide 3 evenly by 2, but this did not seem to throw the
Pythagoreans into a fit. Saying you can’t do a thing simply
because you can’t do it quickly or conveniently or in the way
you initially wanted is just childish. By my method you can
square the circle both physically (to any precision desired) and
abstractly. You just have to be prepared to be logically
consistent in either case.
If all this is true, then why
have we been taught that the circle cannot be squared? It is
because the problem has been used not as a real problem of
geometry or construction, but as an illustration of new theories.
Contemporary mathematicians are most proud of their theory of
transcendental numbers, a theory that is not very old. It is not
surprising that living mathematicians would be more interested in
advertising modern achievements than in fairly presenting a
problem that comes out of the ancient past. Mathematicians enjoy
giving mystical qualities to numbers, and transcendentals bend to
this desire more easily than anything else except complex numbers
or transfinite numbers. By stressing that they are
unconstructible, mathematicians want you to understand that they
are far more esoteric, and therefore worthy of esteem, than
ordinary numbers such as normal people can understand.
Contemporary math has arrived at the furthest reaches of
abstraction, where physicality and constructibility are
considered to be plebeian matters, nearly beneath notice. What a
person can or cannot do with a real tool is mechanics, not
mathematics. What are of interest now are only the equations, and
the equations tell us that π cannot be expressed algebraicly.
That is the fact that must be learned, and if an ancient problem
is misrepresented and much logic bludgeoned to teach it, what
does it matter?
Wikipedia and some other sites
later responded to this paper by adding a sentence like this near
the top of their page:
Approximate
squaring to any given nonperfect accuracy, in contrast, is
possible in a finite number of steps, since there are rational
numbers arbitrarily close to π.
They did not admit
that no figure can
be composed with a compass and a straightedge to perfect
accuracy.
*Most
books admit that the circle can be squared if one is allowed to
roll the given circle. This creates a line of length π and saves
us from having to construct that length with the compass. This is
strictly disallowed, however, since a line created by rolling is
not “constructed.” But the fact that the length π can be
constructed or achieved by any means confirms my arguments above.
Once rolling is allowed and the resulting line is drawn, that
line is understood (and admitted to be, by most modern
mathematicians) exactly
π. It is not an approximation of π, or π with a margin of
error. It is π, here and now and on our diagram, perfect and
complete. The reason its existence in this case confirms my
arguments above is that it is proof of π being determinate. The
end of that drawn or imagined line in the diagram created by
rolling is a mathematical point, which is a precise limit. That
is to say, the line ends abruptly, without any sort of fuzziness
or smearing, quantum or otherwise. This is even clearer when we
are given or achieve a triangle with one side of length π. Both
ends of the line are constrained by the figure, by definition.
Each side of the triangle ends in a discrete point, not in some
bog of an infinite regression. A drawn triangle, which is an
abstract geometic figure, is understood to be a figure that
exists at infinity, where all lines are perfect—with zero
width. If the lines and figures of abstract geometry are at
infinity in this way, then all transcendental lengths must also
be at infinity—that is to say, exhausted.
This means that we don’t need to construct them with compasses
and other real tools, no matter what problem we are working on.
We simply define
all transcendentals as complete, in the same way that we define
all our other abstractions as complete or perfect.
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