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

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