三维球体的贝特朗奇论问题

Three-dimensional sphere Bertrand paradox problem

介绍了平面上圆的随机弦的贝特朗奇论问题,指出了奇论问题具有无穷多个答案,并且答案在一个区间内可连续取值,给出了奇论问题的简单解析,把圆上奇论问题推广到三维空间情形,得出三维球体的贝特朗奇论问题。根据不同的球截面构造方法,给出奇论问题的8种不同解法,得到三维球体贝特朗奇论问题具有无穷多个答案的结论,且除去个别答案之外,其余答案可在一个区间上连续取值。对各种解法进行解析,发现与圆上奇论问题类似,各种解法所确立的随机试验各异是造成一题多种答案的直接原因,而根本原因则是在构造随机弦时对任意性理解的差异。

This paper introduces the Bertrand paradox problem in a random string on the circle, pointing out that thisparadox has infinite answers of continuous values in an intervals and gives a simple analysis to the paradox. The paradoxof circle can be applied into the three-dimensional space to obtain the conclusion of Bertrand paradox in three-dimensionalsphere. According to the different ball section, eight different solutions to this problem are presented in this paper,proving that three-dimensional sphere Bertrand paradox has infinite answers. Despite individual answer, the rest of theanswers can be obtained in a continuous interval. Through analyzing all solutions, we can find similar situations withparadox problem. The immediate reason causing a number of answers is the different randomized trial results, while theessential reason is different understanding of the arbitrary concept when constructing the random string.