This document discusses approximating the trisection of an angle using only a ruler and compass. It presents two methods that can trisect an angle with reasonably high accuracy that is unnoticeable to the naked eye. The first method uses a single bisection, while the second method improves on the first by using a double bisection, making it more accurate for angles over 90 degrees. Theoretical calculations show the error is smallest (0.03%) for acute angles and largest (0.73%) for obtuse angles. Therefore, these methods provide a close approximation of trisecting an angle, even though it is mathematically proven to be impossible.

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Mathematical Analysis and Applications
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Title: Approximating the trisection of an angle
Article Type: Regular Article
Section/Category: Miscellaneous
Keywords: trisecting angle; trichotomy; wantzel; diophantine problems
Corresponding Author: Mr. Chris De Corte, Engineer in Physics
Corresponding Author's Institution: KAIZY BVBA
First Author: Chris De Corte
Order of Authors: Chris De Corte
Abstract: 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.

2. Approximating the trisection of an angle
Author: Chris De Corte
KAIZY BVBA
Beekveldstraat 22 bus 1
9300 Aalst
BELGIUM
Cell: +32 495/75.16.40
Cover Letter

3. 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.
*Manuscript

4. 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.

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 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):

6. Figure 3: Trisecting an angle of 30 degrees using method 1
Trisecting an angle of 45 degrees (figure 4):
Figure 4: Trisecting an angle of 45 degrees using method 1
Trisecting an angle of 60 degrees (figure 5):

7. Figure 5: Trisecting an angle of 60 degrees using method 1
Trisecting an angle of 90 degrees (figure 6):
Figure 6: Trisecting an angle of 90 degrees using method 1
Only now, the first signs of reduced accuracy become slowly visible by the naked
eye.
Trisecting an angle of 120 degrees (figure 7):

8. Figure 7: Trisecting an angle of 120 degrees using method 1
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:

9. 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.

10. 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):
Figure 9: Trisecting an angle of 150 degrees using method 2
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:

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 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):

12. 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).