Tiling Properties of Drainage Basins and Their Physical Bases

Fractality of drainage basins is shown by their tiling properties with subbasins and interbasin areas. Drainage basin forms projected on the two-dimensional plane have mathematical properties similar to those of one-dimensional quasicrystals. Fractal drainage basins are divided into self-similar subbasins and interbasin areas to the infinitesimal limits of their sizes while one-dimensional quasicrystals are divided into segments of two lengths, namely, shorter ones and longer ones also to their infinitesimal limit. The law of stream numbers is expressed by a recurrence formula consisting of three terms. The total number of segments in the line of a quasicrystal is also given by a recurrence formula. In either case, one of the coefficients of them in the recurrence formula is given with the product of solutions of the quadratic equation and the other one with the sum the solutions. The stability of quasicrystals is discussed by using the concept of the Helmholtz free energy. Fractality of drainage basins is explained based on a statistical thermodynamics regarding potential energy expenditure of water in streams. The statistics to explain fractality of drainage basins are peculiar being different from BE, FD and MB statistics. It should be constructed as the nest of a most probable state in a most probable state in a most probable state and so forth.