Abstract: This paper demonstrates the equivalent relation between the 2ⁿ rule of hierarchy of cities and the rank-size rule of city size distribution. According as the rank-size rule, if the largest city P₁ = 1 , then the size of the kth city by rank P_k will be 1?k. Thus city sizes can be abstracted as a harmonic sequences, {1 k}. Grouping the harmonic sequences into many classes in atop-n down way according to the 2 principle yields a hierarchy of fractions with cascade structure . In this instance , the interclass number ratio is r_f = 2. Thetotal population of the first classis 1, second class, 1 2 + 1 3 ≈0. 8333, thethird class, 1 4 + 1 5 + 1 6 + 1 7 ≈0. 7595 , and so on. If the sequence number of a classis large enough, it will have total population approaching to ln2 ≈0. 6931 in theory . By limit analysis, the mean size ratio of two immediate classes is close to r_p = 2. Accordingly, the fractal dimension of the cascade structure is D= lnr_f lnr_p →1. However, the first several classes depart fromthe scaling range to some extent theoretically. As for the empirical data, the last class always goes beyondthe scalingrange because of undergrowth of small cities andtowns. Therefore, the exponential laws andthe power laws of hierarchy of cities are al ways invalid at the extreme scales, i .e . the very large and small scales. Key words 2 n rule ; rank-size rule ; Zipf’s law; fractal ; hierarchy of cities ; cascade structure ; symmetry

Key words: 2ⁿ rule, rank-size rule, Zipf’s law, fractal, hierarchy of cities, cascade structure, symmetry

摘要： 从城市规模分布的位序-规模法则出发, 推导出城市等级体系的二倍数法则。假定城市体系服从标准位序-规模法则(幂指数为1), 则城市规模可以抽象为一个调和数列。按照二倍数的规则对这个调和数列自上而下分级, 各级数值之和在极限条件下趋于常数ln2。由此证明如下问题: 1)城市位序- 规模法则在极限条件下和平均意义上与二倍数法则数学等价; 2)服从位序-规模法则的等级结构在一定尺度范围内是无标度的, 规模尺度最大的城市理论上不服从规律的约束。上述结论可以进行两个方面的推广:一是逻辑推广, 从二倍数推广到多倍数情形; 二是应用范围推广, 从城市研究领域推广到经济学和 自然科学领域。

关键词: 二倍数法则, 位序-规模法则, Zipf定律, 分形, 城市等级体系, 递阶结构, 对称性