On the exotic fishes given to... geometry

Polyhedral forms are extremely widespread both in animate and inanimate nature. Thus, crystals occur as polyhedra only. Besides, these forms are quite common with various primitive organisms, i.e. icosahedral viruses, radiolaria and algae. Here we discuss the cases of exotic Boxfish and Porcupinefish. The specific morphology of the Boxfish reveals in polygonal osseous blades, covering its body. As for the Porcupinefish, its polyhedral approximation was observed via certain geometrical techniques applied. Namely, their spine bases were considered the Delaunay point (R, r)-systems. Consequently, the respective Dirichlet tiling proved to be quasifullerenes and analogous to the Boxfish morphology. This unexpected geometrical dualism of the two families corroborates their taxonomic affinity within the Tetradontiformes order. The above biometrical method is highly recommended as a means of characterization of the Tetraodontiformes specimens in terms of the Delaunay (R, r)-systems

Combinatorial Space Tiling

The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and some old open problems in this area.Comment: 16 pages; to appear in "Symmetry: Culture and Science

A molecular overlayer with the Fibonacci square grid structure

Quasicrystals differ from conventional crystals and amorphous materials in that they possess long-range order without periodicity. They exhibit orders of rotational symmetry which are forbidden in periodic crystals, such as five-, ten-, and twelve-fold, and their structures can be described with complex aperiodic tilings such as Penrose tilings and Stampfli-Gaehler tilings. Previous theoretical work explored the structure and properties of a hypothetical four-fold symmetric quasicrystal-the so-called Fibonacci square grid. Here, we show an experimental realisation of the Fibonacci square grid structure in a molecular overlayer. Scanning tunnelling microscopy reveals that fullerenes (C ) deposited on the two-fold surface of an icosahedral Al-Pd-Mn quasicrystal selectively adsorb atop Mn atoms, forming a Fibonacci square grid. The site-specific adsorption behaviour offers the potential to generate relatively simple quasicrystalline overlayer structures with tunable physical properties and demonstrates the use of molecules as a surface chemical probe to identify atomic species on similar metallic alloy surfaces