return to homepage Squaring the Circle
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. If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many noms de plume. If you are a Paypal user, there is no fee; so it might be worth your while to become one. Otherwise they will rob us 33 cents for each transaction.
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by Miles Mathis
painting by Jake Baddeley
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) = ∞Σn=1(1/n2) = 1/12 + 1/22 + 1/32 + 1/42... = π2/6
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 10-8cm, 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 10-33cm, which requires us to take the equation out to the 1016th 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 zero-width 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 non-starter. 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 four-vector 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 pre-defined 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 zero-dimension 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?
*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.