In this paper we study two classic methods for the approximate construction of regular polygons by using Mathematica 3.0: the method of Archimedes and the method of Bardin. In the same way, we make a comparative study of the errors of both methods, concluding that the exactness of Bardin's method is higher than the Archimedes' one. Moreover, we improve both methods, by giving the respective algorithms. We also include the coded algorithm in Mathematica 3.0 for the animation of both methods. 1 The method of Archimedes The method of Archimedes is a geometric procedure to divide the circumference in a number n of equal parts in an approximate way, and therefore, it has as it's more immediate application the approximate construction of regular polygons. We should point out that although the authorship of the present algorithm is usually attributed to Archimedes, there is no unanimity in this respect in the scientiic community. 1.1 Description of the algorithm 1) Trace a circumference. 2) Trace the main vertical diameter AB of the circumference (as main diameters we understand the two perpendicular diameters that coincide with the coordinated axes), and divide it from top to bottom in n equal parts, where n is the number of sides of the polygon that we want to draw. 3) Trace two arches center on the points A and B respectively and with radius AB. These circumference arches will cut themselves in a point M with positive coordinate of abscissa. 4) We will call N to the second division, beginning from the top, of the diameter AB, and so we obtain the point C of negative coordinate of abscissa in the circumference, result of the intersection of the straight line joining M with N and the circumference. 5) The segment BC is the side that, taken on the circumference, allows us to draw in an approximate way a regular polygon of n sides for inscription in the circumference.
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