p-Adic morphology, biomorphic structures and number theory

The purpose of this article is to establish a link between number theory and morphology and to justify the possibility of using p-adic arithmetic to model biological forms. The work consists of two parts. In the first part we develop some mathematical technique which is based on the theory of locally compact abelian groups and on the concept of p-adic spectrum of series, sequences and infinite products. In the second part, this technique is applied to some classical series and infinite products in order to reveal their new properties and to construct some biomorphic structures, which arise as continuous images of systems of 2-adic balls (or spheres) when they are mapped into the complex plane. The article also introduces the concept of Ω-classes of natural numbers, which are the 2-adic analogue of the residual classes in modular arithmetic. Their connection with the process of discrete 2-adic diffusion is shown. It is shown that Ω-classes produce biomorphic forms with bilateral symmetry and also induce on the set of natural numbers a countable set of multidimensional symmetries. The paper also introduces the concept of the cellular structure of the set of natural numbers and describes a method for constructing from these cells 2-adic organisms based on the metaphor of biological morphogenesis.