Angle trisection: an
approximate graphical method and a trisector
Nicholas Kampouras
Mechanical Engineer M.Sc.
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Abstract
The trisection of an angle
as a problem was posed by the mathematicians in ancient Greece thousands years
ago. Until today attracts the attention of the mathematicians and all those who
love mathematics. It is proved, that an angle trisection can not be done with
the use of the tools the ancient Greeks meant to do it: a straightedge and a
compass. Various methods have been proposed to solve the problem. Some of them
are “exact” from mathematical point of view and some others are approximate. The
present article deals with a method, which could be called “graphical”, because
is based on the graph of the function f(x) =3x-4x3, which actually
represents the trigonometric relation
The idea behind is to construct a curve which will
help to trisect an angle. It is approximate because to plot a graph of values,
with infinite decimal places, is not an “exact” construction. It is a quite
simple and very accurate method.
This article also deals with building a tool
which can trisect any angle. This tool imitates the operation of two engaged
transmission gears.
1.
INTRODUCTION
Geometry was born thousands
years ago. Babylonians developed elementary geometry [1] in order to fulfill
practical needs of the every day life, such as measuring the area of the
fields. For ancient Egyptians geometry became a “secret knowledge” accessible
only to the priests and nobles. Were the ancient Greeks who converted geometry
into the ‘mother of all sciences” [2]. There were three classical problems in
ancient greek mathematics. Squaring the circle, doubling the cube and
trisecting an angle. In 1837, P. Wantzel proved that an angle can not be
trisected by the use of compass and an unmarked ruler. In fact trisecting an
angle means to solve the equation x3 – 3αx2 – 3x + α = 0
which corresponds to the trigonometric
relation
where tan3θ = α and
tanθ = x.
We can list, in
chronological order, the solutions given to the problem of trisection[3]:
@ Hippias (~430
B.C.) using the special curve Quadratrix.
@ Archimedes (287-212 B.C.) solved the problem in two ways, first with the
use of neusis method and second with the use of spiral curve.
@ Nicomedes (~200
B.C.), according to Proclos he used the conchoids curve to solve the problem.
@ Pappus of Alexandria (3rd century) solved the problem in two
ways as well, first with the use of neusis method and second with the use of
hyperbola curve.
@ Al Nasawi (10th-11th century) Arab mathematician,
using neusis method.
@ Rene Descartes (1596-1650) using parabola curve.
@ Blaise Pascal (1623-1662) using limacon curve.
@ Tomasso Ceva (1648-1737) using “The Cycloid of Ceva” curve.
@ Colin Maclaurin (1698-1746) using Trisectrix curve.
@ Delanges (1783) using “The Trisectrix of Delanges”.
@ Plateau (1826) using a curve he invented for this purpose.
@ Longchamps (1888) using a curve he invented for this purpose.
@ Frank Morley (1899) using Morley’s trisection theorem.
@ D.A. Brooks (2007) [4].
There have been also numerous approximate
solutions. An approximate trisection is described by Steinhaus (Wazewski 1945,
Peterson 1983, Steinhaus 1999). Prof. W. Kahan[5] describes two approximate
methods, one of them uses a nomogram.
It is worth to mention the “Origami” folding technique, the “spring method” developed by Hutcheson[6] and the two special tools, which have been fabricated for such a purpose. Fig.1 shows these instruments, the first one (a) is called Angle Trisector and the second one (b) was invented by A. Pegrassi (1893).
It is worth to mention the “Origami” folding technique, the “spring method” developed by Hutcheson[6] and the two special tools, which have been fabricated for such a purpose. Fig.1 shows these instruments, the first one (a) is called Angle Trisector and the second one (b) was invented by A. Pegrassi (1893).
2 2. THE GRAPHICAL METHOD
Consider
the trigonometric relation
Substitute "sinφ" for “y” and "sin(φ/3)" for “x”. Then we have the function y=f(x) =3x-4x3. The independent variable is x which actually depends
on φ.
For deferent values of φ
we have the values of x and correspondently the values of y according to the above
function.
Step
One
Form a table with
the values of φ, sin(φ/3) and sinφ (the values are limited to
nine decimal places) as follows:
φ
|
sin(φ/3)
|
sinφ
|
0O
|
0
|
0
|
10O
|
0.058144829
|
0.173648178
|
15O
|
0.087155743
|
0.258819045
|
20O
|
0.116092914
|
0.342020143
|
25O
|
0.144931859
|
0.422618262
|
30O
|
0.173648178
|
0.5
|
35O
|
0.202217572
|
0.573576436
|
40O
|
0.230615871
|
0.642787610
|
45O
|
0.258819045
|
0.707106781
|
50O
|
0.286803233
|
0.766044443
|
55O
|
0.314544756
|
0.819152044
|
60O
|
0.342020143
|
0.866025404
|
65O
|
0.369206147
|
0.906307787
|
70O
|
0.396079766
|
0.939692621
|
75O
|
0.422618262
|
0.965925826
|
80O
|
0.448799180
|
0.984807753
|
85O
|
0.474600370
|
0.996194698
|
90O
|
0.5
|
1.0
|
Step
Two
Plot the values
of sin(φ/3) and sinφ in a Cartesian system of X-Y axes and draw the
curve, fig.2. The angle φ
takes the values 0 £φ £π. There is no
need to calculate the values of sinφ for π/2
£φ £π
because sinφ = sin(π-φ) (the drawing has been made using AutoCAD
2008).
Step
Three
Extent
the X axis to the left of the curve. On the extension, draw a semi-circle
centered at an arbitrary point of the extension. The circle radius must be 1.0.
The circle radius must equal the height of the curve fig.3
Step
Four
Draw the angle xOy, which we want to trisect,
as follows: One side of the angle (i.e. Ox) must lay on sin(φ/3) axis. The apex of the angle, point O, must
coincide with the center of semi-circle K, fig.4.
Step
Five
The
side Oy intersects the semi-circle at the point A. From point A we draw a line,
parallel to sin(φ/3) axis. This parallel intersects the curve at
the point B. From point B we draw a line, vertical to sin(φ/3) axis. This vertical intersects the sin(φ/3) axis at point C, fig.5.
Step
Six
With a compass we
take on sinφ axis a segment 0D equal in length to 0C,
0D=0C. From point D we draw a line parallel to sin(φ/3) axis. This parallel intersects the
semi-circle at point E. From point E we draw a line to K=O. The angle EOx is the third part of xOy,
fig.6.
Proof
Fig. 6 could be a
“Proof Without Words”, however a detailed proof is following.
From
the right angled triangle AOF (fig.7), sin(xOy) = sin(AOF) = AF/OA and
since OA = 1 > sin(xOy) = sin(AOF) = AF.
Fig. 6 could be a
“Proof Without Words”, however a detailed proof is following.
From
the right angled triangle AOF (fig.7), sin(xOy) = sin(AOF) = AF/OA and
since OA = 1 =>
sin(xOy) = sin(AOF) = AF
Also it is AF = BC, which means that the length
of the segment BC is equal to the arithmetic value of sin(xOy). We have 0C = 0D and 0D = EG. Therefore the
arithmetic value of sin(φ/3) equals the length of the segment EG. . From the right angled triangle EOG =>
From all the above,
Hence
In
fig. 8 we can see an application of the method. In (a) the angle to trisect
measure xOy = 62.8243452O. the one third of xOy must measure 20.9414484O. Using the
above method the trisection of xOy gives 20.9412067O. There is a difference of
0.0002417O. In (b) xOy is 21.3024555O and the trisection
must give an angle of 7.1008185O. The trisection by the method gives
an angle of 7.1008131O, a difference of 0.0000054O. Both
in small and big angles the graphical method gives quite accurate results.
1 3. ANGLE TRISECTOR
In
engineering, when it comes to power transmission from a given gear wheel with
given rpm, to another gear wheel with reduced rpm, the second wheel must have
bigger radius. This fact can be the basic idea for the
construction of an angle trisector.
The proof is very simple. The length of the arc
AB equals the BC, equals the BD.
1 4. CONCLUSIONS
As already
mentioned an “exact” angle trisection is impossible to be done with a compass
and an unmarked ruler. An approximate
method was described in this article. This method uses the graph of the
trigonometric function
The accuracy of
the method depends on the accuracy of the graph. Using the software “AutoCAD
2008” for the plotting of the graph of the function and using values with nine
decimal places, the obtained accuracy was of the order of four decimal places.
Contrary to the above
described method an “exact” trisection can be achieved with the use of a tool,
which imitates the operation of two engaged transmission gears. Two “engaged”
wheels with radius ratio 1:3 can trisect any angle.
1 5. BIBLIOGRAPHY
[1] Eli Maor, “The
Pythagorean Theorm-A 4,000 year history”, Princeton University Press, (2008), p.29
(in Greek).
[2] G. Loria, Guida allo
Studio della Storia delle Matematiche, Milan 1916.
[3] D. Tsimpourakis, “The
geometry and its workers in Ancient Greece”, ALIEN Publications, Athens 1985,
pp. 193-202 (in Greek).
[4] D.A. Brooks, “A new
method of trisection”, College Math. Journal, 38 (2007), 78-81.
[5] Prof. W. Kahan, Math.
Dept. Univ. of Calif. @ Berkeley, “Approximate Trisection of an Angle” Aug. 23,
2005, www.cs.berkeley.edu/~wkahan/Trisect.pdf
[6] Hutcheson, Mathematics Teacher, vol. 94, NO5,
May 2001, pp 400-405.
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