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Diatoms have attracted the interest of those involved in nanotechnology and the development of this discipline for research and practical application. These microorganisms exhibit self-assembly of silica on the micrometer scale, with parts of diatoms measuring on the nanometer scale. They exhibit a wide variety of shapes and patterns. An interest in diatom development and the geometry and topology of diatom shapes and gross patterns formed the focus of this study. Diatoms may be theoretically created on the basis of conic sections and quadratic surfaces expressed in parametric two-dimensional (2D) and three-dimensional (3D) functions (as sheets). Recreated diatom shapes, surfaces, and gross patterns based on 3D parametric equations are presented. Changes from one form to another are discussed as changes in magnitude or functional element within a given set of 3D equations for sequences that are phylogenetically or taxonomically related. Topological evolution and combinations of diatom shapes and surfaces are explored. An example shows how diatom forms depicted as 2-manifolds can be subjected to topological gluing to create a form whose surface is topologically different from the original. Topological surgery and gluing of diatoms as geometric forms may be useful in hypotheses about creating nanosubstrates and nanopatterned nanomaterials or in other applications of nanotechnology.

Geometry and Topology of Diatom Shape and Surface Morphogenesis for Use in Applications of Nanotechnology