approximating the trisection of an angle

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Approximating the trisection of an angle Author: Chris De Corte


Abstract In this document, we will explain how one can quickly and easily trisect an angle with reasonably high accuracy using only a ruler and a compass. As one will see that the used methods will result in a trisection with unnoticeable error by the naked eye. Key-words Trisection of angle; ruler and compass; trichotomy; Pierre Laurent Wantzel; Diophantine problems. Introduction By reading a mathematical book about unsolved problems [1] and later about the history of algebra [2], we learnt that trisecting an angle is one of the impossible things to do. This was confirmed by our research on the internet [3]. So after a while, we became well aware of Pierre Laurent Wantzel’s proof from 1837 that trisecting an angle is mathematically impossible [4]&[5]. Nevertheless, we felt the urge to try this. At a given point, after countless hours and days, we came to a construction that was so close that we thought that we found it. However after testing for obtuse angles we could mark a small difference. We improved our method but still a very small, almost unnoticeable error was there. We found our results so close to target and the used method so special but simple that we nevertheless wanted to share our finding with the mathematical community.


Methods & Techniques Explanation of method 1 using a single bisection: We refer to figure 1.

Figure 1: overview of method 1

We will trisect the inside angle of 2 lines L1 and L2. We will arrange L2 on the x-axis and the intersection of L1 and L2 at the origin O represented by (0,0). Then, we will bisect the angle by ruler and compass (blue line), which we will call line L3. From the origin O, we will draw a reference circle that can be seen as a circle having unit distance. The purpose of this unit circle is to determine the intersection with line L1, which we will call p1, the intersection with line L2, which we will call p2 and the intersection with line L3, which we will call p3. From the point p3, we will intersect line L3 again with our compass using the same distance as before to obtain point p4. From the point p4, we will draw a new circle (using the same unit distance) and we will call this circle C3.


We will now spread the compass open so that it exactly covers the distance O-p4. This will be double the original distance. We will position our compass now in p1 and draw a new circle with double the radius as before and call this circle C1. We will draw a similar circle from p2 and call this circle C2. We will call the intersection of C1 and C3: i1. We will call the intersection of C2 and C3: i2. We draw a line between O and i1. This line will be on 2/3 of the original angle between L1 and L2. We draw a line between O and i2. This line will be on 1/3 of the original angle between L1 and L2. Some examples: We will now show some examples showing the obtained accuracy. The examples are made using a freeware software called “Live Geometry”. In the examples, We drew the 1/3 and 2/3 lines based on the proper formula’s (y=tan(alpha*2/3)*x and y=tan(alpha*1/3)*x) so that the correctness can be immediately visible. Trisecting an angle of 15 degrees (figure 2):

Figure 2: Trisecting an angle of 15 degrees using method 1


Trisecting an angle of 30 degrees (figure 3):









Only now, the first signs of reduced accuracy become slowly visible by the naked eye.




From now on, the accuracy is not so good any more. So for angles above 90 degrees, we introduce the following enhanced method.


Method 2 using double bisection: Alternative method, especially for angles above 90 degrees, demonstrated on a 90 degrees angle: method 2 using a double bisection: We will refer to figure 8:

Figure 8: overview of method 2

We will trisect the inside angle of 2 lines L1 and L2. We will arrange L2 on the x-axis and the intersection of L1 and L2 at the origin O represented by (0,0). Then, we will bisect the angle by ruler and compass (blue line), which we will call line bisect 1. We will also bisect the angle between line bisect 1 and line L2 by ruler and compass (blue line), which we will call line bisect 2 (also blue line). From the origin O, we will draw a reference circle that can be seen as a circle having unit distance. The purpose of this unit circle is to determine the intersection with line bisect 1, which we will call p1, the intersection with line bisect 2, which we will call p2. From the point p2, we will intersect line bisect 2 again with our compass using the same distance as before to obtain point p4. From the point p4, we will draw a new circle (using the same unit distance) and we will call this circle C3.


We will now spread the compass open so that it exactly covers the distance O-p4. This will be double the original distance. We will position our compass now in p1 and draw a new circle with double the radius as before and call this circle C2. We will call the intersection of C2 and C3: i2. We draw a line between O and i2. This line will be on 1/3 of the original angle between L1 and L2. It is easy to double this angle to become the 2/3 line. In words: the intersection between a circle of radius twice unity on a unity distance on the first bisect with a circle of radius unity on twice the unity distance on the second bisect is a good approximation to a point on an angle of 1/3rd the original angle. Other example: Trisecting an angle of 150 degrees (Figure 9):


As can be seen, the accuracy is again very good.


Results In the following, we theoretically calculate the error in our drawings. We refer to figure 8 as well. Method 2 (using a double bisection) in formulas: Circle C3:

Solving C3 for y:

Circle C2:

Replacing y in the above formula by what we found above:

We will try to solve the above equations for x and y and compare them with x and y on the real trisection line:


Calculating the error for some examples (see table 1):

Table 1: calculating the theoretical error of method 2

Following formulas are used in the above excel (figure 10):

Figure 10: formula’s used in table 1

Out of figure 10, we can see that the error is between 0.03% for 45° angles to 0.73% for 150° angles.


Discussions Pierre Laurent Wantzel might be right that it is impossible to algebraically trisect an angle, our method, using only ruler and compass, can easily trick the novice mathematician of the contrary. Conclusion Trisecting an angle, although theoretically not possible, can be fairly approximated using method 2 above. The accurateness is the highest for acute angles. 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://en.wikipedia.org/wiki/Trisecting_the_angle

[4] http://en.wikipedia.org/wiki/Pierre_Wantzel

[5] Wantzel, P.L., Recherches sur les moyens de reconnaitre si un problème de géométrie

peut se résoudre avec la règle et le compass. Journal de Mathematiques pures et

appliques, Vol. 2, pp.366-372, (1837).

approximating the trisection of an angle

Education

degrees figure

angle ruler

a90 degrees angle

inside angle

overview of method

alternative method

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