Random Polygons and Estimations of π

Wen-Qing Xu; Linlin Meng; Yong Li

doi:10.1515/math-2019-0049

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Random Polygons and Estimations of π

Wen-Qing Xu , Linlin Meng und Yong Li

Veröffentlicht/Copyright: 22. Juni 2019

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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.

Keywords: Archimedean polygon; random polygon; random division; extrapolation; Borel-Cantelli lemma

MSC 2010: Primary 00A05; Secondary 60D05; 65C50

1 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 ℝ² can be traced to Archimedes more than 2000 years ago [1]. Although the lower bound π ≈ 3 and better estimates such as π ≈ 3.125 were known to the Babylonians and the Egyptians as early as 4000 years ago, it was Archimedes who first 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 𝓢_n of a regular n-sided polygon inscribed in the unit circle and the semiperimeter Sn′ 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 𝓢_n and Sn′ 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

1/Sn+1/Sn′=2/S2n′,SnS2n′=S2n2

satisfied by the semiperimeters 𝓢_n = n sin (π/n) and Sn′ = 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 𝓢_n and Sn′ for n = 6, 12, 24, 48, 96 and obtain the famous bounds 223/71 < π < 22/7 (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 flavor to the ancient Archimedean approach, we consider in this paper the problem of approximating π using the semiperimeter 𝓢_n or area 𝓐_n 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 confirmed by the strong convergence results stated in the theorem below.

Theorem 1.1

Given n ≥ 3, let 𝓢_n and 𝓐_n 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 𝓢_n and 𝓐_n 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

E(Sn)=π−π3/n2+O(n−3),E(An)=π−4π3/n2+O(n−3).

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 𝓢_n and 𝓐_n that can greatly improve the accuracy of these approximations.

2 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 ΔOAB, sector OAB and ΔOAD, or somewhat differently, by comparing the lengths of BC, arc AB⌢ , and AD, in a unit circle as shown in Fig. 1 below)

$Figure 1 Comparison of areas and lengths in a unit circle: The areas of △ OAB, sector OAB, and △ OAD equal 12 $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$ sin θ, 12 $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$ θ and 12 $\begin{array}{} \displaystyle \frac{1}{2} \end{array}$ tan θ respectively, hence sin θ = |BC| < θ = | AB⌢ $\begin{array}{} \displaystyle \stackrel{\frown}{AB} \end{array}$| < tan θ = |AD| for all 0 < θ < π/2. Note that in the case of a unit circle, θ measures exactly the length of the subtending arc AB⌢ $\begin{array}{} \displaystyle \stackrel{\frown}{AB} \end{array}$. In general, the angle θ, measured in radians, is defined as the ratio of the length of arc AB⌢ $\begin{array}{} \displaystyle \stackrel{\frown}{AB} \end{array}$ to the radius of the arc, a quantity that is dimensionless and independent of the radius of the arc.$

Figure 1

Comparison of areas and lengths in a unit circle: The areas of △ OAB, sector OAB, and △ OAD equal 12 sin θ, 12 θ and 12 tan θ respectively, hence sin θ = |BC| < θ = | AB⌢ | < tan θ = |AD| for all 0 < θ < π/2. 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 defined 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.

sin⁡θ<θ<tan⁡θ,0<θ<π/2, (1)

it is easy to see that 𝓢_n < π < Sn′ for all n ≥ 3. By further applying the related limit

limθ→0sin⁡θθ=1,

it follows that

limn→∞Sn=limn→∞Sn′=π.

Moreover, since the function (sin x)/x is monotone decreasing on the interval (0, π/2), the sequence {𝓢_n} increases with n. On the other hand, since the function (tan x)/x is monotone increasing for 0 < x < π/2, the sequence { Sn′ } decreases with n. Thus, as n becomes larger, the estimates provided by 𝓢_n < π < Sn′ indeed become more and more accurate. Additionally, we note that while the corresponding areas 𝓐_n and An′ of these Archimedean polygons also provide useful approximations of π, with An=12nsin⁡2πn<Sn and An′=ntan⁡πn=Sn′, 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.

Lemma 2.1

Let θ > 0. Then sin θ < θ, sin⁡θ>θ−13!θ3,sin⁡θ<θ−13!θ3+15!θ5,sin⁡θ>θ−13!θ3+15!θ5−17!θ7 , sin θ < θ – 13!θ3+15!θ5−17!θ7+19!θ9 , etc.

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 – θ³/6 and sin θ < θ – θ³/6 + θ⁵/120 for θ > 0, we can establish the following error estimates for 𝓢_n = n sin (π/n)

π−π36n2<Sn<π−π36n2+π5120n4<πfor all n≥3.

Thus, the approximation error associated with 𝓢_n, an under-estimate of π, is slightly less than, but almost precisely π³/(6n²). On the other hand, for the over-estimate of π given by Sn′ = n tan π/n, by using the monotone Taylor series expansion tan θ=θ+13θ3+215θ5+17315θ7+622835θ9+⋯ for the tangent function, we can obtain

Sn′=π+π33n2+2π515n4+17π7315n6+62π92835n8+⋯≈π+π33n2+2π515n4,

with the approximation error slightly more than π³/(3n²). In particular, for n = 96, we find 𝓢₉₆ – π ≈ – π³/55296 ≈ –5.6 × 10^–4 and S96′ – π ≈ π³/27648 ≈ 1.1 × 10^–3.

It is interesting to note that, as one of the greatest mathematicians of all time, Archimedes was wise enough to have stopped at n = 96, but instead suggested taking the average of 𝓢₉₆ and S96′ 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 difference between the areas of △ OAD and sector OAB, is not the same, but about twice as large as the difference 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 ≈ 2(θ – sin θ) for θ small; consequently, instead of the simple average 12Sn+12Sn′, he would have used the weighted average Xn=23Sn+13Sn′ to produce a significantly more accurate estimate of π . (Not until the seventeenth century was such an improvement first 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

$Figure 2 The approximate 1 : 2 : 3 : 6 ratio for the areas of the four small regions in the trapezoid ACBD separated by AB, arc AB⌢ $\begin{array}{} \displaystyle \stackrel{\frown}{AB} \end{array}$, and tangent line BE. The region bounded by AB⌢ $\begin{array}{} \displaystyle \stackrel{\frown}{AB} \end{array}$, BE and EA has the smallest area, followed by the region bounded by AB and AB⌢ $\begin{array}{} \displaystyle \stackrel{\frown}{AB} \end{array}$, and then △ BED, and then △ ACB.$

Figure 2

The approximate 1 : 2 : 3 : 6 ratio for the areas of the four small regions in the trapezoid ACBD separated by AB, arc AB⌢ , and tangent line BE. The region bounded by AB⌢ , BE and EA has the smallest area, followed by the region bounded by AB and AB⌢ , and then △ BED, and then △ ACB.

Xn=23Sn+13Sn′=π+π520n4+π756n6+7π9960n8+⋯

Thus, even with the modest value of n = 96, this would yield π ≈ 𝓧₉₆ – π⁵/1698693120 with an approximation error of about 1.8 × 10^–7, a historic feat that was first achieved by Chinese mathematician Zu Chongzhi more than 7 centuries later by calculating 𝓢_n with n = 2¹² × 3 = 12, 288!

We conclude this discussion by noting that, based on a similar approximate 1 : 3 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 𝓢_n = 𝓐_2n and 𝓐_n (which may also be viewed as an application of modern extrapolation techniques in numerical analysis [3])

Yn=43Sn−13An=43A2n−13An=π−π530n4+π7252n6−π94320n8+⋯ (2)

and further improvements can be obtained by combining 𝓢_n, Sn′ and 𝓐_n in the form

Zn=25Xn+35Yn=1615Sn+215Sn′−15An=π+π7105n6+π9360n8+⋯

and in numerous more ways by also utilizing earlier values such as Sn/2,Sn/2′,An/2, etc.

3 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 definiteness, 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 𝓢_n and area 𝓐_n of such a random n-gon each converges to π with probability 1, that is, ℙ(𝓢_n → π) = ℙ(𝓐_n → π) = 1. This in turn implies convergence of 𝓢_n → π and 𝓐_n → π in probability and in mean square as well.

Suppose the vertices of such an n-gon are labeled P₀, P₁, …, P_n–1, P_n in counterclockwise direction with θ₀ < θ₁ < ⋯ < θ_n–1 < θ_n = θ₀ + 2π and P_n representing the same point as P₀ on the circle. Here θ_i equals the length of the arc from the fixed reference point (1, 0) to P_i, while θ_i+1 – θ_i gives the length of the arc PiP⌢i+1 on the unit circle. The semiperimeter 𝓢_n and area 𝓐_n of the n-gon are then given by

Sn=∑i=1nsin⁡θi−θi−12,An=12∑i=1nsin⁡(θi−θi−1).

Note that, since sin θ < θ for all θ > 0, again we have 𝓐_n < 𝓢_n < π. In fact, we also have 𝓢_n ≤ n sin πn , An≤12nsin⁡2πn. For 𝓢_n, this follows easily from the inequality sin α+sin⁡β=2sin⁡α+β2cos⁡α−β2≤2sin⁡α+β2 for all 0 ≤ α, β ≤ π. For 𝓐_n, the same argument applies if θ_i – θ_i–1 ≤ π holds for all i; and while this is not true, the exception θ_i – θ_i–1 > π occurs with only one index, say i = i_*, then 𝓐_n ≤ 12 ∑_{i≠i_*} sin (θ_i – θ_i–1) ≤ 12(n−1)sin⁡2π−(θi∗−θi∗−1)n−1≤12(n−1)sin⁡πn−1≤12nsin⁡2πn for all n ≥ 3. Thus, in terms of approximating π through either 𝓢_n or 𝓐_n, the regular n-gon always outperforms a random n-gon. (In an extreme case when all n vertices are highly clustered, both 𝓢_n and 𝓐_n can be nearly 0.) Nevertheless, it turns out, convergence remains true for both 𝓢_n and 𝓐_n; 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 difficulty in establishing the convergence of 𝓢_n → π or 𝓐_n → π as n → ∞ arises from the lack of independence among θ_i – θ_i–1 for 1 ≤ i ≤ n (with their sum being 2π). The key to our proof is to use Lemma 2.1 to establish a tight lower bound for 𝔼(𝓢_n) and 𝔼(𝓐_n) with 𝔼(|𝓢_n – π|) → 0 and 𝔼(|𝓐_n – π|) → 0 sufficiently 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–1 are also identically distributed.

Without loss of generality, we assume θ₀ = 0. To further simplify the calculations below, we also write θ_i = 2πX_i, 0 ≤ i ≤ n so that 0 = X₀ < X₁ < X₂ < ⋯ < X_n–1 < X_n = 1 corresponds to a random division [4, 5, 6] of the unit interval by n – 1 uniformly distributed random points, with the lengths of the resulting n segments X_i – X_i–1 = (2π)^–1 (θ_i – θ_i–1) all identically distributed. Since X₁ = min {X₁, X₂, …, X_n–1}, it follows that, for any 0 < x < 1, ℙ (X₁ > x) = ℙ(X_i > x for all 1 ≤ i ≤ n – 1) = (1 – x)^n–1, and thus the probability density function of X₁, and hence of each X_i – X_i–1, is given by f(x) = (n – 1) (1 – x)^n–2. Consequently,

E(|Xi−Xi−1|k)=(n−1)∫01xk(1−x)n−2dx=(n−1)k!(n−2)!(k+n−1)!=k!(n−1)!(k+n−1)!. (3)

In particular, for k = 1, 2, 3, we have

E(|Xi−Xi−1|)=1n,E(|Xi−Xi−1|2)=2n(n+1),E(|Xi−Xi−1|3)=6n(n+1)(n+2). (4)

We now turn to estimate 𝔼(|𝓢_n – π|). First, by using the inequality sin θ>θ−13!θ3 for all θ > 0, we can easily obtain

|Sn−π|=π−Sn≤π36∑i=1n(Xi−Xi−1)3.

With (4), this yields

E(|Sn−π|)≤π36∑i=1nE(|Xi−Xi−1|3)=π3(n+1)(n+2)→0as n→∞.

Thus, by Markov inequality [7, 8], we have, for any ε > 0,

P(|Sn−π|>ε)≤1εE(|Sn−π|)≤π3(n+1)(n+2)ε→0as n→∞.

This proves 𝓢_n → π in probability as n → ∞. Furthermore, we have

∑n=3∞P(|Sn−π|>ε)≤∑n=3∞π3(n+1)(n+2)ε=π34ε<∞.

By applying Borel-Cantelli lemma [7, 8], we see that |𝓢_n – π| > ε occurs finitely often. This implies 𝓢_n → π with probability 1, that is, ℙ(𝓢_n → π) = 1. Additionally, since |𝓢_n – π| ≤ π, we also have the following mean square convergence of 𝓢_n → π as n → ∞:

E(|Sn−π|2)≤πE(|Sn−π|)=π4(n+1)(n+2)→0as n→∞.

With slight modifications in the calculations above, we can obtain similar convergence results for 𝓐_n:

E(|An−π|)≤4π3(n+1)(n+2),E(|An−π|2)≤4π4(n+1)(n+2),

and for all ε > 0,

P(|An−π|>ε)≤1εE(|An−π|)≤4π3(n+1)(n+2)ε→0as n→∞, ∑n=3∞P(|An−π|>ε)≤∑n=3∞4π3(n+1)(n+2)ε=π3ε<∞.

Similar to (2), we can further show that, the combination Yn=43Sn−13An satisfies

Yn=π−π530∑i=1n(Xi−Xi−1)5+π7252∑i=1n(Xi−Xi−1)7−π94320∑i=1n(Xi−Xi−1)9+⋯,

E(|Yn−π|)≤π530∑i=1nE(|Xi−Xi−1|5)=4π5(n+1)(n+2)(n+3)(n+4)→0as n→∞,

and for any ε > 0,

P(|Yn−π|>ε)≤1εE(|Yn−π|)≤4π5(n+1)(n+2)(n+3)(n+4)ε→0as n→∞,

∑n=3∞P(|Yn−π|>ε)≤∑n=3∞4π5(n+1)(n+2)(n+3)(n+4)ε=π590ε<∞.

Note that while the average approximation error for 𝓨_n is now about 120 times that associated with a regular n-gon, it converges to π much faster than 𝓢_n and 𝓐_n 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 𝓢_n and 𝓐_n with the corresponding semiperimeter and area of a suitably constructed 2n-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 [0, 2π] (or tighter estimates described in Section 2), we can obtain

E(Sn)=n(n−1)∫01(sin⁡πx)(1−x)n−2dx=π+∑k=1∞(−1)kn!(n+2k)!π2k+1=π−π3n2+O(n−3),

E(An)=12n(n−1)∫01(sin⁡2πx)(1−x)n−2dx=π+12∑k=1∞(−1)kn!(n+2k)!(2π)2k+1=π−4π3n2+O(n−3),

E(Yn)=43E(Sn)−13E(An)=π−∑k=2∞(−1)k4k−43n!(n+2k)!π2k+1=π−4π5n4+O(n−5),

or alternatively, by repeatedly using integration by parts, the following finite sum expression

E(Sn)=∑k=1(n−1)/2(−1)k−1n!(n−2k)!1π2k−1for n odd,[3ex]∑k=1n/2(−1)k−1n!(n−2k)!1π2k−1+(−1)n/2−1n!πn−1for n even,

E(An)=12∑k=1⌊(n−1)/2⌋(−1)k−1n!(n−2k)!1(2π)2k−1for all n≥3.

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-defined (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.

Acknowledgement

The authors would like to thank Professors Robert Mena, Kent Merryfield, Shu Wang and the anonymous referees for carefully reading earlier drafts of the paper and providing helpful comments and suggestions for improving the presentation of the paper. Research is supported in part by NSFC (Grant No.11471028, 11831003), Beijing Natural Science Foundation (Grant No.1182004, 1192001, Z180007) and Beijing University of Technology (No. ykj-2018-00110).

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Received: 2018-12-12

Accepted: 2019-04-05

Published Online: 2019-06-22

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https://doi.org/10.1515/math-2019-0049

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Archimedean polygon; random polygon; random division; extrapolation; Borel-Cantelli lemma

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