Efficacy of Numerically Approximating Pi with an N-sided Polygon

Transcription

1 Peter Vu Brewer MAT66 Honors Topic Efficacy of umerically Approximating Pi with an -sided Polygon The quest for precisely finding the irrational number pi has been an endeavor since early human history. One of the essential equations for the formula for circumference and area of a circle is C = πr and A = πr with r as the radius, but how exactly can this value of pi be known in the first place? One possible method used is analyzing the geometry of an n-sided polygon on a circle and generalizing the method to estimate the value of pi numerically without any special functions through infinite series. This method can be used to analyze the accuracy of using a polygon to estimate pi for any arbitrary side number. Basic Geometric Concepts The area of inscribed polygon of n-sides is a useful method of approximating pi as the number of sides as depicted below. As the number of sides increase, the area of the polygon estimates more of the circle, which can be used to approximate pi. Using basic trigonometry and arithmetic, the area of an n-sided polygon can be found, although this quickly becomes tedious in calculating each case because of the different methods required for each type of polygon.

2 Generalized derivation of approximation Instead of using geometry to find the area and perimeter of each n-sided polygon, a generalized method using angles and triangles of the polygon can be used to find the area and perimeter of any n-sided polygon to approximate pi. Suppose a regular n-sided polygon is inscribed or circumscribed. This polygon can be divided into n triangles of equal dimensions with equal subdivision of angles within the center of the circle as shown in the figure on the right. Using this to approximate pi, we would have an equation π = n=0 a n, or 0 = π n=0 a n which would be the difference from pi, which will be useful in finding the error. Inscribed Polygon Area For an inscribed regular polygon, the area of sub-triangle using the center angle is A = 1 absin(c) Given the radius R of the circle and angle θ, or θ = π n A = 1 R sin ( π n ) as the n-triangles are equally divided, For an n-sided polygon, n triangles exist, so in approximating the area of the circle πr n = lim n R sin ( π n ) n π = lim n sin (π n ) Instead of depending on the sine function, the series expansion of sine can be found in order to numerically approximate pi without any special function. The power series of sin(x) is known to be

3 xm+1 ( 1) m (m + 1)! So applying this to the previous equation, m+1 π n (π ( 1)m n ) (m + 1)! n ( 3 ( π n ) (π n ) 3! π = π π3 3n + π5 15n 4 error: π3 3n + π5 15n (π n ) 5! ) 4π7 315x 6 4π7 315x 6 Thus, the expression above is the error for an n sided polygon (n > ) for using the inscribed area of an n-sided polygon. Finding a generalized formula for the error series requires concepts beyond the scope of this paper, so the expression will be left as is. Inscribed Polygon Perimeter To find the perimeter of an n-sided regular polygon, the same method can be used except now the side lengths are calculated. Given the radius R of the circle and angle θ = π, the law of n cosines will be useful to solve for the length of one side s of the polygon: c = a + b ab cos(c) s = R + R R cos(θ) s = R (1 cos ( π n )) Equating the side length to the circular perimeter: πr = lim n ns πr = lim n n R (1 cos ( π n ))

4 πr = R lim n n (1 cos ( π n )) π n (1 cos ( π n )) In order to numerically approximate pi without special functions like cosine, the Maclaurin series for cos(x) is known: Hence, xm ( 1) m (m)! π n ( π m (1 n ) ( 1)m (m)! ) This is not so clean as the previous method as the whole series is within a root. Squaring the equation could give an approximate expression for π and thus the error for n-sides however: π = n 4 π = n π ( n ( )m ( 1)m (m)! ) ) m π ( (1 n ) ( 1)m ) (m)! π = n ( 1 (1 (π n )! π π = n (( n )! 4 (π n ) 4! 4 + (π n ) ) 4! 6 + (π n ) ) 6! ) π = π n π4 3n 3 + π6 45n 5 error: n π 3n 3 + π4 45n 5

5 Area of circumscribed polygon Instead of approaching pi through underestimates of an inscribed polygon, a circumscribed polygon can be used to approximate pi by overestimation. As shown in the diagrams above, the sides of the triangles now extend beyond the circle. In the close-up diagram for one triangle, trigonometry of the half angle can be used to determine side length s. cos θ = R L L = R A = 1 absin (C) cos ( θ ) π n A = 1 L sin(θ) R πr = n cos ( θ sin(θ) ) π = n sin(θ) cos ( θ = n sin ( π n ) ) cos ( π n ) )m+1 ( 1) (π m n (m + 1)! = n ( 1) (π m n )m (m)! ( ( π n ) (π n )3 3! 1 (π n )! + (π n )5 5! + (π n )4 4! + + )

6 The resulting approximation equation is rather messy, resulting in a series over a series even with further attempts at simplification such as the half-angle formula for the cosine term. Additionally, methods of simplifying and combining the series together into one general summation formula are beyond the scope of this paper. However, this formula can still be used to computationally estimate the accuracy of pi for n-sides of a polygon. Perimeter of circumscribed polygon The method is very similar to the perimeter of the inscribed polygon, except the sides are derived differently. tan ( θ ) = s R s = R tan ( θ ) πr = ns πr = nr tan ( θ ) π = n tan ( π n ) However, the pattern for the generalized Maclaurin series representation for tan(x) requires methods and concepts beyond the scope of the paper. Regardless, the first few terms of the series representation is calculated as follows: π n ( π n ) + (π n )3 3 + ( π n ) ( π n ) ( ) π = π + π3 3n + π5 15n π7 315n 6 + error: π3 3n + π5 15n π7 315n 6 + Observing the error series, the error will overestimate pi given n is greater or equal to three as the terms are always positive.

7 Alternatively, the law of cosines method can be tested: cos ( θ ) = R L L = R cos ( θ ) c = a + b ab cos(c) s = L + L L cos(θ) s = L (1 cos ( π n )) = ( cos ( θ ) (1 cos ( π ) n )) R s = R (1 cos (π n )) cos ( π n ) = sec ( π n ) sec (π n ) πr = nr (1 cos (π n )) cos ( π n ) m (1 ( 1) (π m n ) (m)! ) π = n ( 1) (π m n )m (m)! Solving for the side length of a circumscribed polygon gives a very messy equation beyond the scope of the paper to use for series expansion. However, this can still be computationally calculated to see how accurate the approximation of pi is for an n-sided polygon.

8 Convergence to pi and Error Using MATLAB, the cumulative sum for each method can be numerically calculated from n = 3 to n = 50 and plotted on a graph as seen in the figure below. Clearly, each plot approaches pi at a different rate, but they all converge to approximate pi with sufficient terms. n Incr. A Error Inscr. P Error Circ. A Error Circ. P Error < < < < <

9 As seen in the table above, most of the geometric methods become accurate to five decimal places by summing terms. The inscribed methods approach pi from the lower bound, while the circumscribed methods approach pi from the upper bound. ote that pi could be just subtracted from all the data points of each plot, essentially giving the error in approximating pi. Overall, all methods approximated pi for a sufficient amount of terms. However, some methods were too complex and messy to approximate numerically without any special functions like trigonometric functions. The cleanest method was inscribing the polygon area, which nicely allowed to determine the error for arbitrary number of sides n for the polygon.

10 References Weisstein, Eric W. "Maclaurin Series." From MathWorld--A Wolfram Web Resource.