4
the circle game the discovery of π
Besides computing the hypotenuse of a triangle, two other geometric problems seem to arise almost inevitably in any numerate civilization: finding the circumference and area of a circle. In contemporary notation, they are given by the closely related formulas C = 2πr and A = πr2. Here A represents the area of a circle of radius r, and π (read “pi”) is the most famous constant in mathematics, the number 3.1415926535…
The modern formulas tend to obscure the first wonderful fact about pi: the fact that the same constant appears in both formulas. They obscure the fact by making it too obvious. To appreciate what ancient mathematicians had to figure out, we should imagine that there is a number “pi-circumference” defined by the ratio of a circle’s perimeter to its diameter d, and a second number “pi-area” defined by the ratio of a circle’s area to its radius squared. Imagine that you don’t know that these two numbers are equal.
The first completely clear statement that the two problems are related comes, not surprisingly, in ancient Greek mathematics. In the third century BC, Archimedes wrote in his manuscript The Measurement of the Circle:
Proposition 1. The area of any circle is equal to the area of a right triangle in which one of the sides about the right triangle is equal to the radius, and the other to the circumference of the circle.
The irrational number π actually has two different meanings. First, it is the ratio of the area A of any circle to the square of its radius r. (That is, π = A/r2.) Second, it is the ratio of the circumference C of any circle to its diameter, d. (That is, π = C/d = C/2r.) Either one of these statements may be taken as a definition of π, and then the other statement becomes a theorem.
Imagine cutting the circle into many wedges, each one indistinguishable from a triangle, so its area is half the base times the height of the wedge, as in the drawing on the next page. The height of each wedge is the radius of the circle, and the sum of all the bases is (roughly) the circumference of the circle. Thus the combined area of all the wedges is (roughly) half the radius times the circumference, and is also (roughly) equal to the area of the circle. The hard part of Archimedes’ argument was turning the rough equalities into exact equalities. Once this is done, it is fairly easy to show that “pi-circumference” is the same as “pi-area.”
Archimedes’ Proposition 1 has been overshadowed by his Proposition 3, where he proved that π lies between 3 1/7 and 3 10/71. But it is really Proposition 1 that gives birth to the concept of pi. Without it, you have two separate problems: how to compute areas and circumferences of circles. With it, you can replace them with a single problem: how to approximate the number pi. Proposition 3 is merely an elaboration of that theme.
As in the case of the Pythagorean theorem, ancient Chinese mathematicians were not far behind their Greek counterparts, if at all. Already in the Nine Chapters—which may predate Archimedes—we find the following problem: “Given a circular field, the circumference is 181 bu and the diameter 601/3 bu. Tell: what is the area? … Rule: Multiplying half the circumference by the radius yields the area of the circle in [square] bu.” The third sentence (the “Rule”) is nothing more or less than Archimedes’ Proposition 1. Interestingly, the first sentence shows that the anonymous author thought that π = 3, a very primitive approximation.
Above A circle cut into wedges, demonstrating Proposition 1.
HOWEVER, LIU HUI, the third-century commentator on the Nine Chapters, had other ideas. To start with, he pointed out that the ratio of the perimeter of a hexagon to its diameter is equal to 3, and yet the perimeter of a circle is visibly larger than that of a hexagon. So the ancient method, based on π = 3, could not be right. “The difference between a polygon and a circle is just like that between the bow and its chord, which can never coincide.” Liu wrote. “Yet such a tradition has been passed down from generation to generation and no one cares to check it.”
To compute a more precise “circle rate,” his term for pi, Liu pushed out each side of the hexagon, to create a 12-sided polygon, and computed the perimeter of that. Then he repeated this procedure to obtain the perimeter of a 24-sided figure, a 48-sided figure, and a 96-sided figure. He also did the same procedure for a 12-sided polygon drawn outside the circle, a 24-sided polygon drawn outside the circle, and so on. In this way, he shows that:
This is very comparable to Archimedes’ estimate:
Above Liu Hui’s demonstration of his “circle rate.”
Archimedes computed his estimate in exactly the same way—starting with a hexagon and doubling the number of sides until he got to a 96-gon! It is amazing that these two great minds, separated by so many miles and so many years, came up with exactly the same idea. The only reason for the slight difference in their answers is that Liu has made more careful approximations along the way. (After the first step, the perimeters involve square roots, which had to be approximated by rational numbers.)
But Liu, unlike Archimedes, didn’t stop! He adds that the procedure can be continued all the way up to a 3072-sided polygon. He omits the calculations, but gives us the result:
He has gotten four digits of pi correct! Liu was probably the first human being to find this now standard approximation to pi.
Curiously, Liu worked pi out to this accuracy once, but in all of his annotations to the other problems in the Nine Chapters, he employed the simpler approximation π ≈ 3.14. This inconsistency points out something very interesting about Liu’s own psychology. He must have realized that there would be no conceivable use for a more accurate approximation in any practical problem. Unless you have a laser interferometer (which didn’t exist back then), you can’t measure the diameter of a field to four decimal places, and so there is no point in using a “circle rate” with that degree of accuracy.
And yet he worked it out to four decimals anyway! He didn’t need to do it; he just wanted to satisfy his own curiosity. He was only the first of many math geeks (or perhaps more specifically pi geeks) over the centuries, who have pushed the computation of pi to almost unfathomable lengths. Before the computer era, William Shanks computed 707 digits of pi, although he tragically made a mistake on the 527th digit, and all the later digits were wrong. Now, with the advent of computers, the record number of digits has been pushed beyond one trillion.
TO PENETRATE this deep into the mysteries of pi, you need more than the relatively clumsy geometric approach of Archimedes and Liu. Around 1500, an unknown Indian mathematician of the Kerala school (possibly Nilakantha Somayaji or his predecessor Madhava) discovered the exquisite formula:
now known as the Gregory–Leibniz formula after its first European discoverers. Such formulas, relating pi to infinite sums of simple fractions, became much easier to derive with the invention of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the late 1600s. A personal favorite, proved by Leonhard Euler in 1734, is the amazing equation:
The symbol π for the “circle rate” was also introduced about this time (in 1706 by William Jones), and popularized by Euler.
Take a moment to reflect on the beauty of these formulas. These equations reveal that the number pi is not merely a geometric concept. Three of the great tributaries of mathematics merge in these formulas: geometry (the number pi), arithmetic (the sequence of odd numbers, and the sequence of squares 12, 22, 32, …), and analysis of the infinite (in this case, infinite sums). Archimedes would have been flabbergasted to see formulas like these. Liu would have been speechless. And they would have gone straight out to buy a book on calculus and learn this wonderful new art.
And yet there are even deeper levels to the number pi. It is irrational—a fact that eluded ancient mathematicians, although they must have suspected it. Johann Lambert proved the irrationality of pi in 1761. A century later, Ferdinand Lindemann (in 1882) proved the more subtle fact that pi is transcendental, a kind of souped-up version of irrationality.‡ Lindemann’s theorem resolved the ancient problem of squaring the circle, posed by the
ancient Greeks: Is it possible, with only basic geometric operations, to draw a square whose area is the same as a given circle? A positive answer to this question—a method for squaring the circle—would have made the number pi more accessible to them. Alas, it was not to be. Transcendental ratios cannot be constructed with a ruler and compasses.
Above A mosaic illustrating the pi symbol, laid into the floor outside the mathematics building at the Technische Universität Berlin.
Even today, there are facts we still do not know about pi, and discoveries presumably still waiting to be made. As recently as 1995, three mathematicians—David Bailey, Peter Borwein, and David Plouffe—discovered a brand-new formula for pi that may deserve to be etched on the same mountaintop as Leibniz’s and Euler’s. It is the first self-repairing formula for pi, in the sense that if you make a mistake at the 527th place, it doesn’t invalidate your later calculations. However, there is a catch. The self-correcting property is only true if you write pi in hexadecimal (base 16) arithmetic, as computers do.§ It won’t work in ordinary (decimal) notation. So if God created the integers, and God created pi, then perhaps God is actually a computer.
* * *
‡ A number is transcendental if it cannot be expressed as the solution to any polynomial equation with rational coefficients. For instance, √2 is not transcendental, because it solves the equation x2 = 2.
§ In hexadecimal notation, π = 3.243F6A8885A308D3… The letters “A” through “F” stand for the numbers 10 through 15, which are single digits in base 16.
5
from zeno’s paradoxes to the idea of infinity
The city of Elea, in the present-day province of Salerno, Italy, was home to two noted philosophers who spanned the period between Pythagoras and Socrates. Parmenides, the elder of the two, was noted for his beliefs that all things are one and that the world we perceive is different from the world of reality—a viewpoint that would strongly influence Plato’s philosophy.
Parmenides’ student, Zeno, is noted not so much for any particular beliefs as for a style of debate that Aristotle called dialectic, in which you argue against your opponent’s beliefs rather than in favor of your own. Zeno would take his opponent’s belief as a premise and try to prove logically that the premise led to an absurdity. His arguments are usually called “paradoxes” because they seem to refute very commonly-held beliefs. For example, suppose that you believe that it is possible to move from point A to point B. Before you can reach point B, Zeno argues, you must have gone halfway to B. Before you can get halfway to B, you must have gone half of that distance (or a quarter of the distance to B), and so on. In other words, you must have completed an infinite number of motions before you can even travel the tiniest fraction of the distance from point A to point B! Clearly, Zeno says, this is absurd. Therefore, motion is impossible.
A second paradox is called Achilles and the tortoise. If you believe in motion, Zeno says, then you must surely believe that the swift Achilles can catch up with a slow-moving tortoise. But he argued that if Achilles runs to where the tortoise is now, the tortoise will have already moved a few steps forward. If Achilles runs to that place, the tortoise will have moved forward once again, and so on. Again, Achilles must complete an infinite number of tasks in a finite time, and that (at least to Zeno) is clearly absurd.
The ellipsis (…) means that the sum is to be taken ad infinitum, not stopped after a finite number of steps. More formally, you can get as close as you want to 2 if you add up a sufficiently large (finite) number of terms in this sum.
To the modern-day mathematician, Zeno’s paradoxes are harmless. In fact, they are a rather perceptive description of what continuous motion is all about. Let’s say that Achilles is going twice as fast as the tortoise, and the tortoise starts with a 1-yard head start. After 1 second, Achilles has traveled 1 yard and the tortoise has traveled 1/2 yard. (This is one fast tortoise!) After 1 + 1/2 seconds, Achilles has traveled (1 + 1/2) yards and the tortoise has traveled (1/2 + 1/4) yards. Where will Achilles and the tortoise be after n of these steps? And how much time will elapse? Working through the sums, the amount of time elapsed is just shy of 2 seconds. In fact, it is:
seconds, or more simply seconds
Achillles has traveled just less than 2 yards, in fact yards
The tortoise has traveled only half that far, i.e. yards
But it had a 1-yard head start, and if we add its head start to the distance it has traveled, we see it is yards ahead of Achilles’ starting point. Because the tortoise is closer to the 2-yard mark than Achilles is. Therefore Zeno is correct in asserting that after seconds Achilles is still behind the tortoise.
So we know where Achilles and the tortoise are just a split second before 2 seconds, and a split split second after that, and a split split split second after that … But no man, not even Zeno, can stop time. Eventually the stopwatch will reach 2 seconds. Where will Achilles and the tortoise be then? The answer is that they will both be at the point they have been getting closer and closer to—2 yards beyond Achilles’ starting point. Modern mathematicians call this “taking the limit” as n approaches infinity. Both of the terms 1/2n and 1/2n+1 approach zero, and so disappear in the limit. After 2 seconds, Achilles has traveled 2 yards, the tortoise has traveled 1 yard; Achilles has caught up.
Where did Zeno go wrong? First, he started to mathematize the problem but did not finish the job: he left out important information, namely the time elapsed. Second, and more importantly, he and the other ancient Greeks were still sufficiently uneasy about the concept of infinity that they could not take the limit. That is, they could not go from the finite sum:
to the infinite sum:
But they tried so hard! And they came so close! Just how close becomes apparent when you read Archimedes’ Quadrature of the Parabola, written about two centuries after Zeno.
In this document, written as a letter to a fellow mathematician, Dositheus, upon the death of a mutual friend named Conon, Archimedes writes: “While I grieved for the loss not only of a friend but of an admirable mathematician, I set myself the task of communicating to you, as I had intended to send to Conon, a certain geometrical theorem which had not been investigated before but has now been investigated by me, and which I first discovered by means of mechanics and then exhibited by means of geometry. Now some of the earlier geometers tried to prove it possible to find a rectilineal area equal to a given circle … but I am not aware that any one of my predecessors has attempted to square the segment bounded by a straight line and a section of a right-angled cone [i.e., a parabola].” He goes on to say that he has proved that any such region has area equal to 4/3 times the area of an inscribed triangle whose height is the same as the height of the parabolic region.
Below right Zeno (ca. 334–262 BC). This engraving first appeared in The History of Philosophy by Thomas Stanley, published in London in 1656.
Quotes such as this give us insight into the character of Archimedes—another math geek. The best way he can think to console Conon’s friend is to send him the proof of a new mathematical theorem! Also, notice the reference to squaring the circle—a subject that Archimedes had some experience with. Archimedes is not able to square the circle but he is able to “square” or rectify a different curved region, a far from obvious accomplishment. And finally, notice he draws a curious distinction between “discovering” the theorem by means of mechanics and “exhibiting” (or proving) it by means of geometry.
AS IT TURNS OUT, the method that Archimedes used to estimate the area of a circle works much better for a parabola. And this time there is nothing approximate about it: Archimedes says that the area of the parabola is exactly 4/3 that of the inscribed triangle. To prove this, he pushes out two of the sides of the triangle T, creating a four-sided figure that more closely approximates the parabola. Then he pushes out those sides, creating an eight-sided figure, and so on. And at each step, he shows, he adds one quarter of the area that was added in the previous step. So if we take th
e area of the initial triangle as 1, then the area of the four-sided figure is 1 + 1/4. The area of the eight-sided figure is 1 + 1/4 + 1/16. Continuing in this fashion, after n steps, he has an extremely close approximation to the parabola, whose total area is:
Above The area of the parabola is equal to 4/3 of the triangle.
This strongly resembles the sum seen when analyzing Zeno’s paradox, only involving powers of 4 rather than 2. Next, Archimedes shows that adding 1/3 of the last term to this finite sum always gives a total of exactly 4/3:
Remember that the ancient Greeks were uncomfortable with numerical proofs, so Archimedes had to prove this geometrically, as shown below. Suppose the L-shaped region labeled A has area 1. Then the area of the large square containing it is 4/3 (because the large square has four equal quadrants, only three of which are contained in region A). The large square can be filled out, or “exhausted,” by a shrinking sequence of L-shaped regions, plus one small leftover square in the lower right-hand corner (labeled D). The total area of these pieces is the left-hand side of the equation above. Thus the left-hand side and the right-hand side are equal.
Left The area of the large square is 4/3 the area of the region labeled A.
What a wonderful argument! But notice that Archimedes stops after n steps; he doesn’t let the procedure run “on to infinity.” A modern mathematician would have no qualms about taking this step. The parabola is “exhausted” by successive triangles, just as the square is exhausted by the successive L-shaped figures. Taking the limit as the number n approaches infinity, we would conclude that the parabola’s area equals the square’s area, which we showed was 4/3. In other words, we would use the infinite sum:
The Universe in Zero Words Page 4