Branching Structures in Nature and Human Societies

Abstract

expressions of human relations and communication networks. Some of these examples appear in this review article.As a mathematical framework developed since many years we have the method developed by a geologist Horton (1945), who found so-called Horton's law in the tree-type river branching structures, which is introduced briefly in Sec. 2. This method has a great advantage in a sense that the geometrical properties of tree-type structures can be expressed in terms a single parameter. On the other hand, the network-type branching systems have been often treated successfully by scientists from various fields, but they were not based on a simple method similar to that of Horton. The present author has once proposed a method to treat network-type system for leaf veins and road systems, which was included in a monograph by the present author (Takaki, 1978) and not published as a scientific paper. It is introduced in Sec. 4.Of course, the framework of analysis of branching systems is not limited to the Horton's method, and some remarkable examples are introduced in the following sections. In particular, an application of the topology (one of mathematical fields) is made by a pathological scientist Shimizu (1992) for analysis of 3-dimensional (3D) network-type structures of blood vessels in human liver along with a topological concept called "Betti number".Here, it is expected that introductions of various method and concepts would give a larger scope of branching systems, which would stimulate further development of studies in future. Horton's Law for River Structures and Its Derivations Horton's lawA river made of branching streams has a shape belonging to the category of tree-type. The tree structure of riv- Branching Structures in Nature and Human Societies Ryuji TakakiTokyo University of Agriculture and Technology (emeritus professor), 2-23-12 Yuigahama, Kamakura, Kanagawa 248-0014, Japan E-mail address: jr.takaki@iris.ocn.ne.jp (Received July 19, 2015; Accepted March 25, 2016) Several examples of branching structures in the nature, social structures and the human body are introduced, and results of their analyses are given. It is shown that the Horton's law, which is confirmed for river branching structures, is satisfied also in variety of branching structures in the nature and human societies. It is suggested that these structures are constructed owing to mechanisms similar to that for river structure. As for the branching structures in human body some trials are introduced to construct them numerically by the use of mathematical models. The object of this review article is to show that the analyses of branching forms are interesting topics as the science of forms.

Forms in Economics and Business Administration

“…On the second day, Professor Takeshi Yoshida from Aoyama Gakuin University School of Business Administration also gave a presentation as a guest lecturer (Yoshida, 2013), and Professor Hiroki Takada from Fukui University Graduate School of Engineering gave a presentation as a special lecturer (Takada, 2013). In addition, some society members also gave insightful reports during general presentation sessions (Suzuki et al, 2013;Takaki, 2013).…”