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# Approximating the trisection of an angle

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In this document, I 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.

I am well aware of Pierre Laurent Wantzel’s proof from 1837 that trisecting an angle is mathematically impossible.

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### Approximating the trisection of an angle

1. 1. Elsevier Editorial System(tm) for Journal ofMathematical Analysis and ApplicationsManuscript DraftManuscript Number:Title: Approximating the trisection of an angleArticle Type: Regular ArticleSection/Category: MiscellaneousKeywords: trisecting angle; trichotomy; wantzel; diophantine problemsCorresponding Author: Mr. Chris De Corte, Engineer in PhysicsCorresponding Authors Institution: KAIZY BVBAFirst Author: Chris De CorteOrder of Authors: Chris De CorteAbstract: In this document, I will explain how one can quickly and easilytrisect an angle with reasonably high accuracy using only a ruler and acompass.As one will see that the used methods will result in a trisection withunnoticeable error by the naked eye.I am well aware of Pierre Laurent Wantzels proof from 1837 thattrisecting an angle is mathematically impossible.
2. 2. Approximating the trisection of an angleAuthor: Chris De CorteKAIZY BVBABeekveldstraat 22 bus 19300 AalstBELGIUMCell: +32 495/75.16.40Cover Letter
3. 3. AbstractIn this document, we will explain how one can quickly and easily trisect an anglewith reasonably high accuracy using only a ruler and a compass.As one will see that the used methods will result in a trisection with unnoticeableerror by the naked eye.Key-wordsTrisection of angle; ruler and compass; trichotomy; Pierre Laurent Wantzel;Diophantine problems.IntroductionBy reading a mathematical book about unsolved problems [1] and later about thehistory of algebra [2], we learnt that trisecting an angle is one of the impossiblethings 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 from1837 that trisecting an angle is mathematically impossible [4]&[5].Nevertheless, we felt the urge to try this. At a given point, after countless hoursand days, we came to a construction that was so close that we thought that wefound it. However after testing for obtuse angles we could mark a smalldifference. We improved our method but still a very small, almost unnoticeableerror was there.We found our results so close to target and the used method so special butsimple that we nevertheless wanted to share our finding with the mathematicalcommunity.Methods & TechniquesExplanation of method 1 using a single bisection:We refer to figure 1.*Manuscript
4. 4. Figure 1: overview of method 1We 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 Orepresented by (0,0).Then, we will bisect the angle by ruler and compass (blue line), which we will callline L3.From the origin O, we will draw a reference circle that can be seen as a circlehaving unit distance.The purpose of this unit circle is to determine the intersection with line L1, whichwe will call p1, the intersection with line L2, which we will call p2 and theintersection with line L3, which we will call p3.From the point p3, we will intersect line L3 again with our compass using thesame distance as before to obtain point p4.From the point p4, we will draw a new circle (using the same unit distance) andwe 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 theradius as before and call this circle C1. We will draw a similar circle from p2 andcall this circle C2.We will call the intersection of C1 and C3: i1.
5. 5. 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 anglebetween L1 and L2.We draw a line between O and i2. This line will be on 1/3 of the original anglebetween 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 beimmediately visible.Trisecting an angle of 15 degrees (figure 2):Figure 2: Trisecting an angle of 15 degrees using method 1Trisecting an angle of 30 degrees (figure 3):
6. 6. Figure 3: Trisecting an angle of 30 degrees using method 1Trisecting an angle of 45 degrees (figure 4):Figure 4: Trisecting an angle of 45 degrees using method 1Trisecting an angle of 60 degrees (figure 5):
7. 7. Figure 5: Trisecting an angle of 60 degrees using method 1Trisecting an angle of 90 degrees (figure 6):Figure 6: Trisecting an angle of 90 degrees using method 1Only now, the first signs of reduced accuracy become slowly visible by the nakedeye.Trisecting an angle of 120 degrees (figure 7):
8. 8. Figure 7: Trisecting an angle of 120 degrees using method 1From 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 a90 degrees angle: method 2 using a double bisection:We will refer to figure 8:
9. 9. Figure 8: overview of method 2We 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 Orepresented by (0,0).Then, we will bisect the angle by ruler and compass (blue line), which we will callline bisect 1. We will also bisect the angle between line bisect 1 and line L2 byruler 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 circlehaving 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 usingthe same distance as before to obtain point p4.From the point p4, we will draw a new circle (using the same unit distance) andwe 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 theradius as before and call this circle C2.We will call the intersection of C2 and C3: i2.
10. 10. We draw a line between O and i2. This line will be on 1/3 of the original anglebetween 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 unitydistance on the first bisect with a circle of radius unity on twice the unity distanceon the second bisect is a good approximation to a point on an angle of 1/3rd theoriginal angle.Other example:Trisecting an angle of 150 degrees (Figure 9):Figure 9: Trisecting an angle of 150 degrees using method 2As can be seen, the accuracy is again very good.ResultsIn 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:
11. 11. 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 andy on the real trisection line:Calculating the error for some examples (see table 1):Table 1: calculating the theoretical error of method 2Following formulas are used in the above excel (figure 10):
12. 12. Figure 10: formula’s used in table 1Out of figure 10, we can see that the error is between 0.03% for 45° angles to0.73% for 150° angles.DiscussionsPierre Laurent Wantzel might be right that it is impossible to algebraically trisectan angle, our method, using only ruler and compass, can easily trick the novicemathematician of the contrary.ConclusionTrisecting an angle, although theoretically not possible, can be fairlyapproximated using method 2 above. The accurateness is the highest for acuteangles.AcknowledgementsI would like to thank this publisher, his professional staff and his volunteers for allthe effort they take in reading all the papers coming to them and especially Iwould 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 countlesshours I spend behind my desk.References[1] De zeven grootste raadsels van de wiskunde;Alex van den Brandhof, Rolandvan 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 géométriepeut se résoudre avec la règle et le compass. Journal de Mathematiques pures etappliques, Vol. 2, pp.366-372, (1837).