Growth and Form – Volume 4

The aim of Growth and Form is to publish papers on a wide range of models from applied mathematics for modeling biological phenomena and growth. Since its inception we have published several interesting papers, with very different perspectives. Volume 4 contains three articles, one review article and two research articles. The review article by Edoardo de Tommasi and Alessandra Rogato discusses the link between shape and function to understand the evolutionary success of diatoms, unicellular microalgae which contribute massively to primary production and carbon sequestration on a global scale. The peculiar feature of diatoms is the frustule, a porous silica shell which encloses the protoplasm, which is characterized by the presence of regular patterns of micro-and nano-pores in an ultra-structured architecture. The diatom frustule seems to be involved in mechanical protection of the cell, sorting of nutrients from noxious agents, gas exchange and efficient coupling with solar radiation. The discussion on frustule morphogenesis and optical functions is linked to Gielis transformations as an elegant and efficient tool to describe its geometry. The article by Matthew He deals with the genetic code on the one hand and a variety of mathematical phenomena on the other. The biological system divides the genetic four-letter alphabet of DNA and RNA into various three pairs of letters. The article investigates the geometric shapes and forms associated with these kinds of genetic code equivalences and shows that each equivalence has its own geometric shape and form. The geometric properties include attracting fixed point, repelling fixed point, basin of attractions, Julia sets and corresponding Mandelbrot sets, the golden ratio, ring ratio and unity ratio matrices associated with three kinds of genetic code equivalences.