Approximations in drawing π and squaring the circle

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Approximations in drawing π and squaring the circle

Author: Chris De Corte


Abstract In this document, I will explain how one can theoretically draw π (pi), up to any desired accuracy, using only a ruler and a compass. Afterward, I will show the easiest way to approximate the squaring of a circle. I am well aware that “Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) published in 1882 his proof that π (pi) is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients”. The reader is warned that following this, it should not be possible to draw π hence to square the circle. Key-words Lindemann–Weierstrass, pi (π), squaring, circle, subset-sum problem, NP-complete, NP-hard Introduction By reading a mathematical book about unsolved problems [1] and later about the history of algebra [2], we learnt that squaring the circle is one of the impossible things to do. This is because one cannot draw pi (π). In order for a line to be constructible, the value need to be a root of a polynomial. However, π is transcendental, and satisfies no polynomial equation over the rationals [3]. Reading this was enough to be challenged to nevertheless give it a try.


Methods & Techniques Method used for drawing π: Using simple geometry, it is easy to draw the following basic starting angles: 72° (pentagram), 60°, 45°. Let's call them αi. The above basic angles can be split n times in 2, each time using a simple bisection, resulting in n bisections to the new angle αi /2

n. From all these angles, it is also easy to project the radius of a unit circle onto the x- or y-axis to obtain the cosine, sinus and even the tangent. So, we will list a limited list of all these possible angles in a spreadsheet software and also calculate the respective trigonometric values of them. Then we will sort the values from high to low and manually start selecting the best fit values to construct our approximation of π up to the desired number of digits accuracy. The result can be seen in table 1. We only attempted to construct 3 different approximations of π up to 8 digits accuracy. Because of possible rounding inaccuracies in the spreadsheet software, we didn't make attempts of higher accuracy than 8 digits but this should theoretically not be a problem with other software.


table 1: constructing π from projections of constructible angles


Out of the above spreadsheet, we can deduct following approximations for π: Approximation 1:

Approximation 2:

Approximation 3:


Squaring the circle Using the above method to determine and draw π, it is clear that there will be an infinite number of ways to square any circle up to a desired accuracy. However, during my attempts, I discovered a particularly easy way to approximate this by using the following method: Draw a first circle of size unity (1.00) as a start. Using two bisections of this agreed unity to obtain the length of 0.25, draw also a second circle of unity times 1.25 from the same origin. Draw a +45° and -45° line through the origin of both the circles. The 4 intersections with the second circle will serve as the corners of the square that will be the approximation of the squared first circle. The error will only be 0.26% (=(1.2533141-1.25)/1.2533141) which will be as good as unnoticeable by the naked eye. This is because to be correct, the second circle should have had radius sqrt(π/2) which is 1.2533141 while we approximated it to be 1.25.


Results The examples to theoretically draw π only uses simple angle bisections on a limited list of basic angles followed by addition and subtraction of projections. No multiplication, division, squares, roots or nested roots are used of the projections. So, many more complicated scenario's are possible. I think that it is clear from the spreadsheet and from the simple example that:

1. approximations to any more digits accuracy is theoretically possible. For each extra digit in accuracy that is wanted in π, we will always find a combination of tang, cos or sin (no matter close to 0° or 90° angles) that will make us able to construct the desired accuracy of π.

2. An infinite number of combinations will be possible to approximate π to the desired accuracy in this way (as we can always increase one of the coefficients of the last approximation and recalculate the other coefficients so that the new approximation again achieves the desired accuracy).

Discussions


Two open question remain:

1. Considering that π is transcendental. But also most of the used sinus, cosine and tangents are also transcendental. Does the proof of Ferdinand von Lindemann also include the use of (x=) transcendentals to come to a new transcendental (π)? I wouldn't think so because otherwise, the polynomial π = 1 * π would proof the contrary. Suppose this reasoning is correct, how could one be sure that there does not exist a lucky combination that will work out true exactly?

2. If the answer to the above question that there is definitively no solution that results in an exact value for π remains valid, then, which formula (with combinations of addition, subtraction, division and multiplication of nested roots and squares of basic angles and powers of bisected angles) will come most close to π?

I think the finding of the answer to either of the above 2 questions has similarities to the subset-sum problem (given a set of integers, is there a non-empty subset whose sum is zero? ). The subset-sum problem is a NP-complete problem. Our problem however looks like harder to solve (since there are many more operations possible than sum only and they could be nested) and hence this problem could be an NP-hard problem.

Conclusion Drawing π up to any agreed number of digits is theoretically possible using only a compass and a ruler. Squaring a circle can be easily approximated with the shown method. Conjectures:

1. the number of ways to draw π up to an agreed number of digits is infinite 2. the listing of the set of ways and finding the best approximation is an NP

hard problem


Acknowledgements I would like to thank this publisher, his professional staff and his volunteers for all the effort they take in reading all the papers coming to them and especially I would like to thank this reader for reading my paper till the end. I would like to thank my wife for keeping the faith in my work during the countless hours I spend behind my desk. References [1] De zeven grootste raadsels van de wiskunde;Alex van den Brandhof, Roland van der Veen, Jan van de Craats, Barry Koren; Uitgeverij Bert Bakker [2] Unknown Quantity; John Derbyshire; Atlantic Books [3] http://math.stackexchange.com/questions/121453/can-a-line-of-lengths-be-drawn-

with-a-compass-and-straight-edge

[4] https://en.wikipedia.org/wiki/Squaring_the_circle

[5] Geometric Constructions; Lesley Lamphier; Iowa State University [6] Geometrical Constructions; Mathematics is not a Spectator Sport; Phillips G., Springer [7] Fundamentals of Geometry; Oleg A. Belyaev [8] http://en.wikipedia.org/wiki/P_versus_NP_problem [9] http://en.wikipedia.org/wiki/Subset_sum_problem [10] http://en.wikipedia.org/wiki/Pentagon [11] http://mathworld.wolfram.com/AngleTrisection.html

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Approximations in drawing π and squaring the circle