点空间分析——分维与均匀度

POINT SPATIAL ANALYSES——FRACTAL DIMENSION AND UNIFORM INDEX

对有限空间内的点集定义了独占圆、独占线和独占球。通过对有限点集均匀性的研究,抽象出了均匀度的定义。对分形集定义了度量映射,度量映射产生的点集的均匀度与分维之间有换算关系,可以说均匀性与复杂性是密不可分的。均匀度是对点集格局的一种测度,它描述的是点集的空间关系,而不是点的“多少”,有限点集的均匀度可以取到[01]区间的任何实数,这是它与测度和分维的区别。本文得到两个有趣的常数:一位无限循环小数产生的点集的均匀度为0.1,随机格局的均匀度的数学期望为1/x=0.318。可见,均匀度将空间复杂性转移到了点集均匀性上,它为复杂性的研究打开了新的窗口。

The concepts, monopolized circle, monopolized line and monopolized sphere were put forward in this paper. The uniform index was put forward based on the study of the uniformity about finity point set. New concept measuring mapping was defined to fractal set, the fractal dimension of the fractal set can be transformed to the uniform index of the point set created by measuring mapping, it is shown that the complexity and uniformity are tight. The uniform index is a measure of point set pattern, which describes the spatial relationship in the point set, but the number. The uniform index can be any real number in interval , which is different from mathematical measure and fractal dimension. Two very interesting numbers, the uniform index of 1 digit infinite repeating decimal and random pattern are 0.1 and 1/x=0.318.