CC BY
32
A STUDY OF PARTICULAR METHODS FOR THE APPROXIMATE CONSTRUCTION OF SOME REGULAR POLYGONS BY USING MATHEMATICA 3.0
J.A. Aledo (Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain)
J. C. Cortes (Department of Mathematics, LE.S. Bonifacio Sotos, 02200 Casas Ibanez, Albacete, Spain) F.L. Pelayo (Corresponding Author. Department of Computer Science, University of Castilla-La Mancha, 02071 Albacete, Spain. Tel.: 34-967599200 ext:2410; fax: 34-967599224; e-mail: fpelayo@info-ab.uclm.es)
constructed by the corresponding particular algorithm.
In the other hand, let d{Pi,P2) be the euclidean distance between the points Pi, P2 G R2, and C[P,r] = C[(a, 6),r] the circumference centered on P = (a, 6) and radius r. In particular C[(0,0), 1] is
2) Determinate the point Pi of negative ordinate, which is result of the intersection of C[(0,0), 1] and
’Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain, t Department of Mathematics, I.E.S. Bonifacio Sotos, 02200 Casas Ibdnez, Albacete, Spain.
* Corresponding Author. Department of Computer Science, University of Castilla-La Mancha, 02071 Albacete, Spain. Tel.: 34-967599200 ext:2410; fax: 34-967599224; e-mail: fpelayo@info-ab.uclm.es
In this paper we study particular methods for the approximate construction of the regular pentagon, heptagon, enneagon, decagon and undecagon by using Mathematica 3.0. In the same way, we make a study of the errors of those methods whose construction is not exact, and we compare them with those obtained in the general methods of Archimedes and Bardin. We also include the coded algorithm in Mathematica 3.0 for the animation of each method.
1 Preliminaries
Let us LEn denote the exact longitude on the side of the regular polygon of n sides inscribed in a circumference of radius 1, this is
-n = 2 sin ( — ) Vn > 3,
and Ln the longitude on the side of the regular polygon of n sides inscribed in the same circumference
the goniometric circumference and C[P, d(M, N)] the circumference centered on P and radius equal to the distance between M and N.
2 Construction of the regular pentagon
2.1 Description of the algorithm
1) Trace the circumference C[(0,0), 1] and its principal diameters AB and CD (as main diameters we understand the two perpendicular diameters that coincide with the coordinated axes).
C|(1,0),1].
4) Determinate the point P3 of negative abscissa, which is result of the intersection of C[Pi,d(Pi, P2)] and the diameter CD.
5) The distance L5 = d{B, P3) is the side of the regular pentagon that we want to draw.
D
Figure 1: Construction of the pentagon
2.2 Analysis of the error
Let us prove, by using Mathematica 3.0, that above algorithm is exact, this is, that L5 = LE5. The points Pi and P2 can be founded by means of
resulting
Solve[{x2 + y2 == 1, (x - l)2 + y2 == 1}, {x, y}] Solve[{x2 + (y + l)2 == 1, x2 + y2 == 1}, {x, y}]
and
To calculate P3 we previously need compute the distance between Pi and P2, that it can be obtain by using
distance[{a_,b_}, {c_,d_}] := \J(c- a)2 + (d - b)2 dis(Pl, P2) := distance[{1/2, ->/3/2}, {->/3/2, -1/2}]
Solve[{(x - 1/2)2 + (y + \/3/2)2 == dis(Pl, P2)2,y == 0}, {x,y}]
outputting
Finally, the side L5 can be calculated in the way
Lado5 := distance[{0, —l}, {(1 - yf&)/2, 0}] that is equal to LEs as can be proved by means of the following algorithm
ExactSide[n_] := 2 * Sin — j Abs[ExactSide[5] — Lado5]
3 Construction of the regular heptagon
3.1 Description of the algorithm
1) Trace the circumference C[(0,0), 1] and its principal diameters AB and CD.
2) Determinate the point Pi and P2, result of the intersection of C[(0,0), 1] and C[(0, — 1), 1].
3) Determinate the point P3, result of the intersection of the straight line joining Pi with P2 and the diameter AB.
4) The distance L7 = d(Pi,Ps) is the side of the regular heptagon that we want to draw.
N
A
Figure 2: Construction of the heptagon
3.2 Analysis of the error
Let us start determinating the points Pi and P2
Solve[{x2 + y2 == l,x2 + (y + l)2 == 1}, {x,y}]
obtaining
and therefore
Hence, we can calculate Lj by means of
Lado7 := distance[{0, —1/2}, {(—>/3/2, —1/2}]
resulting the value L7 = \/3/2.
We will obtain an estimate of the precision of the method starting from the relative error (LEj — Lj)/LEj. The reason why we don’t consider the error taking the absolute value of the difference (LE7 — L7), is that we want to know if the value L7 obtained by means of the algorithm approaches for excess
or for defect to the exact longitude LE7. In particular, if the relative error is positive then the approach is for defect, else it is for excess.
The code
RelativeErrorHeptagon := (ExactSide[7] — Lado7)/ExactSide[7]
produces the value 2.00754 • 10 3, i.e., the error of the method is roughly 0.2%.
4 Construction of the regular enneagon
4.1 Description of the algorithm
1) Trace the circumference C[(0,0), 1] and its principal diameters AB and CD.
2) Determinate the point P\ of positive abscissa , which is result of the intersection of C[(0, 0), 1] and
3) Determinate the point P2 of positive abscissa . which is result of the intersection of C[(0,1), d(A, Pi)] and the diameter CD.
4) Determinate the point P3 of negative abscissa , which is result of the intersection of C[P2, d(P2, B)] and the diameter CD.
5) The distance Lg = d(C, P3) is the side of the regular enneagon that we want to draw.
C[(0,-1),1].
N
A
B
Figure 3: Construction of the enneagon
4.2 Analysis of the error
Let us calculate Pi by means of the code
*
Solve[{x2 + y2 == 1, x2 + (y + l)2 == 1}, {x, y}]
resulting
To determinate P2 can be used the following algorithm
dis(A,Pl) := distancia[{0, l}, {\/3/2,-1/2}] Solve[{x2 + (y - l)2 == dis(A, PI)2, y == 0}, {x, y}]
which outputs P2 = {n/2,0}.
On the other hand, we can find P3 in the way
dis(P2,B) := distance[{\/2,0}, {0, — 1}]
Solve[{(x - \/2)2 + y2 == dis(P2, B)2, y == 0}, {x, y}|
obtaining P3 = {\/2 — s/3,0}.
Finally, the code
Lado9 := distance[{—1,0}, {V2 — \/3,0}]
allows us to calculate Lg = 1 — \/3 + y/2.
In this case the relative error of the method is the output generated by the input
RelativeErrorEnneagon := (ExactSide[9] - Lado9)/ExactSide[9]
resulting the value 2.74477 • 10-3, i.e, the error of the method is roughly 0.3%.
5 Construction of the regular decagon
5.1 Description of the algorithm
1) Trace the circumference £[(0,0), 1] and its principal diameters AB and CD.
2) Determinate the point Pi and P2, result of the intersection of C[(0,0), 1] and C[(0, — 1), 1].
3) Determinate the point P3, result of the intersection of the straight line joining P\ with P2 and tl diameter AB.
4) Determinate the point P4 of negative abscissa, result of the intersection of the straight line joinir P3 with C and C[P3,d(P3, B)\
5) The distance L10 = d(P4,C') is the side of the regular decagon that we want to draw.
N
Figure 4: Construction of the decagon
5.2 Analysis of the error
We will check that the above algorithm allows us to construct the regular decagon in an exact way.
Observe that the points Pi. P2 and P3 coincide with its homologous ones in the case of the construction of the heptagon. To calculate P4 we can use
Solve[{x2 + (y + 1/2)2 == 1/4. y == -(1 2) * (x + 1)}. {x.y}]
obtaining
Therefore. Lio can be determinated by means of
LadolO := distance[{ — 1. 0}. | ~ — (—5 4- \/5)|]
resulting the value 1,0= (-1 + \/5)/2.
To prove that this algorithm is exact, it is enough to check that the following value vanishes
Abs[ExactSide[10] - LadolO]
6 Construction of the regular undecagon
6.1 Description of the algorithm
1) Trace the circumference C((0,0), 1] and its principal diameters AB and CD.
2) Determinate the point P\ of positive abscissa . which is result of the intersection of C[(0,0), 1] and C[(0,-1),1].
3) Determinate the point P2, result of the intersection of the straight line joining Pi with A and the diameter CD.
4) The distance Ln = d(Pi,P2) is the side of the regular undecagon that we want to draw.
N
Figure 5: Construction of the undecagon
6.2 Analysis of the error
The point Pi coincide with the point P2 in the subsection 3.2, this is, P, = (^3/2,-1/2). Now, P2 can be calculating by using
Solve[{y == -\/3x + 1, y == 0}, {x, y}]
obtaining P2 = (\/3/3,0).
Finally we can obtain L\\ by means of the code
Ladoll := distance[{\/3/3, 0}, {\/3/2, —1/2}]
which provides the value
L n —
1 (-V3 y/3
4 V 3 2
Therefore, the relative error of the method
ErrorRelativoUndecagono := (LadoExacto[ll] - Ladoll)/LadoExacto[ll]
results —2.46424 • 10-2, i.e., in this occasion we approach for excess and the error is greater than the error of before cases.
7 Comparison with the general methods of Archimedes and of Bardin
Table 1 compares the relative errors of the particular algorithms for the construction of the pentagon, enneagon and undecagon with the relative errors that result of the layout of these polygons by using the general methods of Archimedes and of Bardin.
n Rel. Error of the particular algorithm Rel. error Archimedes Rel. Error Bardin
7 2.00754 • 10~3 — 1.62426 • 10-3 2.86033 • 10-4
9 2.74477- 10"J -6.65798- 10"3 1.45450- 10“3
11 —2.46424 • 10“2 — 1.24955 • 10-2 1.78029-10~3
Table 1: Comparison of the relative errors
It is curious that a general method like Bardin’s one allows us to obtain better results than the corresponding particular methods. However, the particular construction of the enneagon is better than that obtained by the method of Archimedes.
8 Animation of these methods by using Mathematica 3.0
We have made one program for each method, which outputs the respective guided construction (animation) of the regular polygon.
These programs will be shown at IMACS-ACA’99.
References
[1] J.A. Aledo, J.C. Cortes y F.L. Pelayo, Estudio de dos metodos clasicos para la construction aproxi-mada de poh'gonos regulares usando Mathematica 3.0, Preprint.
[2] C.B. Boyer, Historia de las Matematicas, primera edition, Ed. Alianza Universidad Textos, Madrid, (1987), 195-204.
[3j J.C. Cortes Lopez, Estudio matematico del trazado general de polfgonos regulares, Epsilon 39 (1997), 149-158.
[4] F.J. Rodriguez de Abajo y V. Alvarez, Dibujo tecnico, EM. Donostiarra, (1985), 55-57.
[5] S. Wolfram, Mathematica, a system for doing mathematics by computer, Ed. Adison-Wesley (1991).
[6] S. Wolfram, The Mathematica book, Wolfram Media, Inc (1996).
TEACHING EFFICIENT MATHEMATICS
Gennadi Malaschonok, Natalia Malaschonok (Tambov State University, 392622 Tambov, Russia e-mail: malaschonok@math-iu.tambov.su)
The new generation of computer algebra systems, that have developed during the last 5 years, is characterized by thorough solving the problems and user friendliness. Solving mathematical problems that needed the numerous tedious calculations becomes now the fascinating work. The penetration of mathematics into various a spheres of knowledge swiftly intensifies.
The XXI century would be the age of new chemical and biological technologies, moreover, it would be the age ot new mathematical technologies. Contemporary systems of Computer Algebra are primary among such technologies. New mathematical technologies require an alteration of all systems of mathematical education.
The education of professional mathematicians must include a compulsory course titled ’’New mathematical technologies”. This course should acquaint the fledgling professionals with the main technological tools and their usage, along with some test cases of their successful implementation.
The teaching of non-mathematicians is also to be changed radically. It should be based on intensive usage of the mathematical methods, especially with their new contemporary possibilities. The future specialist must have an opportunity to appreciate the potentialities of mathematics. So his mathematical education must train him for that. The high-effective computer systems have to be the necessary component of a such training. The mathematical education must consist of two parts. The first part is teaching to the language of mathematics. The second part is teaching to use the mathematical instrument, that is formulation of applied problem, solving, analysis, interpretation of solutions with the help of mathematical technologies.
The task of mathematicians is the creation of universal computer mathematical systems and special mathematical packets for each applied sphere. This is the way the system Mathematica acts. The mathematical packet for applied sphere is a laboratory for special experiments, where the results would be received with extreme rapidness and low expenses.
