Random Polygons and Estimations of π

Abstract In this paper, we study the approximation of π through the semiperimeter or area of a random n-sided polygon inscribed in a unit circle in ℝ2. We show that, with probability 1, the approximation error goes to 0 as n → ∞, and is roughly sextupled when compared with the classical Archimedean approach of using a regular n-sided polygon. By combining both the semiperimeter and area of these random inscribed polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.


Introduction
The classical approach to estimate π, the ratio of the circumference of a circle to its diameter, based on the semiperimeter (or area) of regular polygons inscribed in or circumscribed about a unit circle in R can be traced to Archimedes more than 2000 years ago [1]. Although the lower bound π ≈ and better estimates such as π ≈ .
were known to the Babylonians and the Egyptians as early as 4000 years ago, it was Archimedes who rst used the polygonal method to calculate π to any desired degree of accuracy. On the one hand, Archimedes correctly recognized that π lies between the semiperimeter Sn of a regular n-sided polygon inscribed in the unit circle and the semiperimeter S n of a similar regular n-gon circumscribed about the circle; On the other hand, being a master of the method of exhaustion, he certainly knew that as n gets larger and larger, both Sn and S n get closer and closer to π. Furthermore, with the doubling of the sides of the polygons, Archimedes also discovered the following harmonic-geometric-mean relations /Sn + /S n = /S n , SnS n = S n satis ed by the semiperimeters Sn = n sin(π/n) and S n = n tan(π/n) of the respective regular n-sided polygons inscribed in and circumscribed about the unit circle. These recurrence relations allowed him to actually compute Sn and S n for n = , , , , and obtain the famous bounds / < π < / (and provided essentially the only tool to obtain more accurate estimates of π for later mathematicians until about the seventeenth century).
To introduce some modern avor to the ancient Archimedean approach, we consider in this paper the problem of approximating π using the semiperimeter Sn or area An of an n-sided random polygon inscribed in the unit circle. For simplicity, we assume that all vertices are independently and uniformly distributed on the circle. By connecting these vertices consecutively, we then obtain a random polygon inscribed in the unit circle. Note that although such random polygons will rarely be regular (when the vertices happen to be all equally spaced on the circle), it is intuitively clear that, as n becomes large, these random vertices tend to spread out and become "evenly" distributed on the circle so that the semiperimeter or area of the circle may still be well approximated by the corresponding semiperimeter or area of the inscribed random polygon. This is con rmed by the strong convergence results stated in the theorem below. Theorem 1.1. Given n ≥ , let Sn and An be the semiperimeter and area of a random inscribed polygon generated by n independent points uniformly distributed on the unit circle. Then, with probability 1, both Sn and An converge to π as n → ∞.
Note that Theorem 1.1 improves on the weak convergence results previously obtained by Bélisle [2]. In fact, for n large, we can also obtain the error estimates Thus, compared with a regular n-gon which happens to minimize the approximation error, on average, the approximation error is roughly sextupled when a random n-gon is used. Additionally, we will also show that, for both Archimedean and our random approximations of π, by applying extrapolation type techniques [3], it is possible to construct some simple linear combinations of Sn and An that can greatly improve the accuracy of these approximations.

Basic convergence estimates for the Archimedean approximations of π
By using the following well-known elementary estimates (which can be derived, for example, by comparing the areas of ∆OAB, sector OAB and ∆OAD, or somewhat di erently, by comparing the lengths of BC, arc AB, and AD, in a unit circle as shown in Fig. 1 below) sin θ < θ < tan θ, it is easy to see that Sn < π < S n for all n ≥ . By further applying the related limit Moreover, since the function (sin x)/x is monotone decreasing on the interval ( , π/ ), the sequence {Sn} increases with n. On the other hand, since the function (tan x)/x is monotone increasing for < x < π/ , the sequence {S n } decreases with n. Thus, as n becomes larger, the estimates provided by Sn < π < S n indeed become more and more accurate. Additionally, we note that while the corresponding areas An and A n of these Archimedean polygons also provide useful approximations of π, with An = n sin π n < Sn and A n = n tan π n = S n , there seems to be no clear advantage in doing so-something Archimedes might have reasonably concluded.
The following lemma provides some improved higher-order estimates for the sine function and will be useful for deriving error estimates for various approximations of π.
Note that these inequalities correspond precisely to estimates given by the partial sums of the alternating Taylor series of the sine function. By using sin θ > θ − θ / and sin θ < θ − θ / + θ / for θ > , we can tan θ respectively, hence sin θ = |BC| < θ = | AB| < tan θ = |AD| for all < θ < π/ . Note that in the case of a unit circle, θ measures exactly the length of the subtending arc AB. In general, the angle θ, measured in radians, is de ned as the ratio of the length of arc AB to the radius of the arc, a quantity that is dimensionless and independent of the radius of the arc.
establish the following error estimates for Sn = n sin(π/n) Thus, the approximation error associated with Sn, an under-estimate of π, is slightly less than, but almost precisely π /( n ). On the other hand, for the over-estimate of π given by S n = n tan π/n, by using the monotone Taylor series expansion tan θ = θ + θ + θ + θ + θ + · · · for the tangent function, we can obtain with the approximation error slightly more than π /( n ). In particular, for n = , we nd S − π ≈ −π / ≈ − . × − and S − π ≈ π / ≈ . × − . It is interesting to note that, as one of the greatest mathematicians of all time, Archimedes was wise enough to have stopped at n = , but instead suggested taking the average of S and S for a better approximation of π. However, had it ever occurred to him that for θ = π/n small, while sin θ < θ < tan θ, that is, the area of sector OAB is "sandwiched" between those of OAB and OAD, the di erence between the areas of OAD and sector OAB, is not the same, but about twice as large as the di erence between the areas of sector OAB and OAB (see Fig. 2 below for a more complete comparison), he would have arrived at the more useful estimate tan θ − θ ≈ (θ − sin θ) for θ small; consequently, instead of the simple average Sn + S n , he would have used the weighted average Xn = Sn + S n to produce a signi cantly more accurate estimate of π. (Not until the seventeenth century was such an improvement rst pointed out and then rigorously proved by Dutch mathematicians Snellius and Huygens respectively [1].) From the Taylor expansions for sin θ and tan θ, we see that Xn = Sn + S n = π + π n + π n + π n + · · · Thus, even with the modest value of n = , this would yield π ≈ X − π / with an approximation error of about . × − , a historic feat that was rst achieved by Chinese mathematician Zu Chongzhi more than 7 centuries later by calculating Sn with n = × = , ! We conclude this discussion by noting that, based on a similar approximate : ratio between the area bounded by AB and AB and the area of ACB, a slightly more accurate estimate for π can be achieved by using the following combination of Sn = A n and An (which may also be viewed as an application of modern extrapolation techniques in numerical analysis [3]) and further improvements can be obtained by combining Sn, S n and An in the form Zn = Xn + Yn = Sn + S n − An = π + π n + π n + · · · and in numerous more ways by also utilizing earlier values such as S n/ , S n/ , A n/ , etc.

Approximation of π through the semiperimeter or area of a random cyclic n-gon
We now turn to the related but more interesting problem of approximating π through the semiperimeter or area of a randomly selected n-gon inscribed in a unit circle, adding another modern twist to Archimedes' ancient approach. For de niteness, we assume that the vertices of the n-gon are independently and uniformly distributed on the circle. Our main goal is to show that, as n → ∞, the semiperimeter Sn and area An of such a random n-gon each converges to π with probability 1, that is, P(Sn → π) = P(An → π) = . This in turn implies convergence of Sn → π and An → π in probability and in mean square as well. Suppose the vertices of such an n-gon are labeled P , P , . . . , P n− , Pn in counterclockwise direction with θ < θ < · · · < θ n− < θn = θ + π and Pn representing the same point as P on the circle. Here θ i equals the length of the arc from the xed reference point ( , ) to P i , while θ i+ − θ i gives the length of the arc P i P i+ on the unit circle. The semiperimeter Sn and area An of the n-gon are then given by Note that, since sin θ < θ for all θ > , again we have An < Sn < π. In fact, we also have Sn ≤ n sin π n , An ≤ n sin π n . For Sn, this follows easily from the inequality sin α + sin β = sin α+β cos α−β ≤ sin α+β for all ≤ α, β ≤ π. For An, the same argument applies if θ i − θ i− ≤ π holds for all i; and while this is not true, the exception θ i − θ i− > π occurs with only one index, say i = i * , then An ≤ i≠ i * sin(θ i − θ i− ) ≤ (n − ) sin π−(θ i * −θ i * − ) n− ≤ (n − ) sin π n− ≤ n sin π n for all n ≥ . Thus, in terms of approximating π through either Sn or An, the regular n-gon always outperforms a random n-gon. (In an extreme case when all n vertices are highly clustered, both Sn and An can be nearly 0.) Nevertheless, it turns out, convergence remains true for both Sn and An; and in fact, it is not at all a bad idea to use the semiperimeter or area of a random n-gon to approximate π since a typical approximation error is only about 6 times that of the regular n-gon.
Before we proceed, we mention that the main di culty in establishing the convergence of Sn → π or An → π as n → ∞ arises from the lack of independence among θ i − θ i− for ≤ i ≤ n (with their sum being π). The key to our proof is to use Lemma 2.1 to establish a tight lower bound for E(Sn) and E(An) with E(|Sn − π|) → and E(|An − π|) → su ciently fast as n → ∞. In particular, we will exploit the symmetry (all vertices are independent and identically distributed) which implies that all θ i − θ i− are also identically distributed.
Without loss of generality, we assume θ = . To further simplify the calculations below, we also write θ i = πX i , ≤ i ≤ n so that = X < X < X < · · · < X n− < Xn = corresponds to a random division [4][5][6] of the unit interval by n − uniformly distributed random points, with the lengths of the resulting n segments X i − X i− = ( π) − (θ i − θ i− ) all identically distributed. Since X = min{X , X , . . . , X n− }, it follows that, for any < x < , P(X > x) = P(X i > x for all ≤ i ≤ n − ) = ( − x) n− , and thus the probability density function of X , and hence of each In particular, for k = , , , we have We now turn to estimate E(|Sn − π|). First, by using the inequality sin θ > θ − ! θ for all θ > , we can easily obtain With (4), this yields Thus, by Markov inequality [7,8], we have, for any ε > , This proves Sn → π in probability as n → ∞. Furthermore, we have By applying Borel-Cantelli lemma [7,8], we see that |Sn − π| > ε occurs nitely often. This implies Sn → π with probability 1, that is, P(Sn → π) = . Additionally, since |Sn − π| ≤ π, we also have the following mean square convergence of Sn → π as n → ∞: With slight modi cations in the calculations above, we can obtain similar convergence results for An: , E(|An − π| ) ≤ π (n + )(n + ) , and for all ε > , Similar to (2), we can further show that, the combination Yn = Sn − An satis es and for any ε > , Note that while the average approximation error for Yn is now about times that associated with a regular n-gon, it converges to π much faster than Sn and An for large n. It should be clear that, with the doubling of the sides of such a random n-gon, further extrapolation improvements may be obtained [9] by combining Sn and An with the corresponding semiperimeter and area of a suitably constructed n-sided random polygon inscribed in the unit circle. In fact, besides the above mentioned strong convergence results, central limit theorem type (weak) convergence estimates also hold for these random approximations of π [2,9].
On the other hand, by using (3) and the uniform and absolute convergence of the Taylor series of sine function on the interval [ , π] (or tighter estimates described in Section 2), we can obtain E(Sn) = n(n − ) (sin πx)( − x) n− dx = π + ∞ k= (− ) k n! (n + k)! π k+ = π − π n + O(n − ), E(An) = n(n − ) (sin πx)( − x) n− dx = π + ∞ k= (− ) k n! (n + k)! ( π) k+ = π − π n + O(n − ), or alternatively, by repeatedly using integration by parts, the following nite sum expression for n odd, We mention that, while only random inscribed polygons are considered in this paper, most of our convergence results actually also hold for random circumscribing polygons [10] that are tangent to the circle at each of the prescribed random points. However, unlike the classical Archimedean case, such a circumscribing random polygon is not always well-de ned (when all random points fall on a semicircle), and even if it exists, its semiperimeter or area can still be unbounded. Finally, similar convergence results also hold for certain random cyclic polygons whose vertices are no longer independently and uniformly distributed on the circle. We refer to [10,11] for details.