海岸线的长度依赖于测量尺度,因此没有确定的测量结果:尺度越小,测量的长度值越大。国境线的测量也具有类似的尺度依赖性。这种性质与分形性质有关。分形的本质是标度对称性,而对称性意味着某种不可观测量。由于标度对称性的存在,一些地理现象的基本测度(如长度、面积)无法准确计算。中国、美国等国家的国土面积大小都没有确切的数据:在版图没有任何变动的情况下,不同时期、不同机构的测量结果不一致,有时甚至大相径庭。本文借助分形模型和标度对称思想探讨这一问题。假定一个国家包括漫长的海岸线,则其国土可以分解两部分:主体部分可以用Koch岛模型刻画,岛屿部分则可以用Pareto分布描述。这两部分叠加的结果是,国土总面积随着测量尺度的减少而逐步增加。因此,中国政府可以正式公布国土面积的最新测量数据,而不必继续担心由此引发的国际政治争端。

What on earth is the size of China’s land area in total? No person can answer this question.Indeed,this paper tries to discuss the scale dependency of geographical area measurement rather than the total area of Chinese land.In theory,a country bordering on oceans can be divided into two parts.The main land can be modeled with Koch’s island,and the islands and islets in the marginal seas can be described with the Pareto distribution.The Koch island model suggests that the boundary line of the land area is a fractal line,while Pareto’s law suggests that the size distribution of islands and islets in an ocean is of scaling invariance.A coastline,consisting of bays and headlands,has irregularity at a certain degree,which was discussed by B.B.Mandelbrot in his fractal works.On the other hand,an archipelago is made up of many islands and islets.Zoom in on a smaller portion and it,in turn,will be composed of smaller islands and islets.The distribution of islands and islets does not change with distance.In technical terms,the archipelago is said to be of self-similarity at different scales,that is,it is a fractal set of random phenomena.
A fractal is based on a form of symmetry with scale invariance under contraction or dilation.The scale invariance is known as dilation symmetry or scaling symmetry.Symmetry implies some kind of immeasurable quantity.Because of scaling symmetry,the length of coastline,the area of a country’s land,and so on,cannot be well and truly measured.Measuring the area of a country from space will give a lower total than measuring it from 100 meters away,because of the extra detail at a larger scale.As the scale gets finer,more details(smaller islands or islets) can be picked up.The inevitable conclusion is that the total area of the country appears to increase indefinitely.This is just the scale-dependency of geographical measurements.The essence of land area is similar to that of the coastline length.The result(area or length) depends on the scale at which you make the measurement.Therefore,the land area data of different sources is not always comparable due to different sizes of yard measure,different scales of digital maps and different resolutions of geographical images.