On Murray Law for Optimal Branching Ratio

Abstract

Tree-like branching networks are widespread in nature and have found wide applications in engineering, where Murray’s law is generally adopted to optimally design tree-like systems, but it may become invalid in some cases. Here we give an energy approach to the analysis of the law and re-find Li–Yu’s law for the optimal ratio of the square root of 2 with a suitable constraint. When the cross-section of each branch is considered as a fractal pattern, a modified Murray’s law is obtained, which includes the original Murray’s law for a Peano-like pore and Li–Yu’s law for cylindrical branches, furthermore a useful relationship between the diameter and length of each hierarchy is obtained, which is contrary to the tree-like fractal patterns, and the new hierarchy is named as “fractal Murray tree”, which also has many potential applications in science, engineering, social science and economics. This paper is intended to serve as a foundation for further research into the fractal Murray tree and its applications in various fields.