圆的平方问题的近似解

Тагир Пшуков, Tagir Pshukov, Мурат Османович Мамчуев, M. Mamchuev
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摘要

众所周知,圆的平方(构成一个与给定圆面积相同的正方形的问题),以及立方体和角三等分的重复,是用罗经和直尺构造的最著名的无法解决的几何问题之一。将圆的平方问题的求解简化为对圆进行矫直,即构造与圆长度相等的线段,其不可解性与圆周率字符的超越性有关。本文证明了Christian Huygens定理中的一个极限情况,该定理通过圆内切的正多边形的周长来估计圆的周长。在此基础上,提出并证明了一种求解圆平方问题的近似方法,该方法可以连续地构造任意精确的问题解。我们将近似一个半径为给定圆半径倍数的圆的弧,借助于与收缩的弦平行的线段,然后按要求的次数增加或减少这个线段,这样得到的线段的长度将大约等于给定圆的周长的一半。我们所考虑的圆弧越小,近似精度就越高。但实际工具的可能性是有限的,不适合太小和太大的绘图比例。为了解决这一问题,提出了一种缩放近似算法,该算法只需要增加(或减少)绘图碎片,就可以使所有时间都停留在相同尺寸的薄片内。也许这种方法将对其他结构有用,包括那些需要非常大或反之亦然的非常小的图纸尺寸的结构。
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Approximate Solution for Squaring the Circle Problem
It is known that squaring the circle (the problem consisting in construction of a square with the same area as a given circle), together with duplication of cube and angle trisection, is one of the most famous unsolv able problems of constructive geometry for construction with compass and straightedge. The solution of squaring the circle problem is reduced to the straightening of the circle, that is, to the construction of a segment equal in length to the circle, and its insolvability is connected with the pi character transcendence. In this paper, the limiting case of one of Christian Huygens theorems, which establishes an estimate for length of circumference of a circle through perimeters of regular polygons inscribed in circle and circumscribed about it, is proved. On this basis has been proposed and justified an approximate method for squaring the circle problem solving, which allows consistently construct arbitrarily exact solutions of the problem. We will approximate an arc of a circle whose radius is a multiple of the given circle’s radius, with the help of a segment which is parallel to a shrinking it chord, and then will increase or decrease this segment in the required number of times, so that the resulting segment’s length would be approximately equal to half of the given circle’s circumference. The approximation accuracy will be the higher the smaller arc of the circle we will consider. But possibilities of real tools are limited, and not suitable for both too small and too large drawing scales. In order to cope with this problem, an algorithm for scaled approximation has been proposed, in which it is sufficient to increase (or reduce) the drawing fragment, so that all the time sta y within the sheet of the same size. Perhaps this approach will be useful for other constructions, including the exact ones, where it is necessary to come to very large or vice versa very small drawings’ dimensions.
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