Tiling mechanisms of the compound eye through geometrical tessellation

Tilling patterns are observed in many biological structures. Hexagonal tilling, commonly observed in the compound eyes of wild-type Drosophila, is dominant in nature; this dominance can probably be attributed to physical restrictions such as structural robustness, minimal boundary length, and space filling efficiency. Surprisingly, tetragonal tiling patterns are also observed in some Drosophila small eye mutants and aquatic crustaceans. Herein, geometrical tessellation is shown to determine the ommatidial tiling patterns. In small eye mutants, the hexagonal pattern is transformed into a tetragonal pattern as the relative positions of neighboring ommatidia are stretched along the dorsal-ventral axis. Hence, the regular distribution of ommatidia and their uniform growth collectively play an essential role in the establishment of tetragonal and hexagonal tiling patterns in compound eyes.

BUFFON’S COIN AND NEEDLE PROBLEMS FOR THE SNUB HEXAGONAL TILING

In this paper we consider the snub hexagonal tiling of the plane ($(3^4, 6)$ Archimedean tiling) and compute the probability that a random circle or a random segment intersects a side of the tiling.

A decagonal quasicrystal with rhombic and hexagonal tiles decorated with icosahedral structural units

The structure of a decagonal quasicrystal in the Zn58Mg40Y2 (at.%) alloy was studied using electron diffraction and atomic resolution Z-contrast imaging techniques. This stable Frank–Kasper Zn–Mg–Y decagonal quasicrystal has an atomic structure which can be modeled with a rhombic/hexagonal tiling decorated with icosahedral units at each vertex. No perfect decagonal clusters were observed in the Zn–Mg–Y decagonal quasicrystal, which differs from the Zn–Mg–Dy decagonal crystal with the same space group P10/mmm. Y atoms occupy the center of `dented decagon' motifs consisting of three fat rhombic and two flattened hexagonal tiles. About 75% of fat rhombic tiles are arranged in groups of five forming star motifs, while the others connect with each other in a `zigzag' configuration. This decagonal quasicrystal has a composition of Zn68.3Mg29.1Y2.6 (at.%) with a valence electron concentration (e/a) of about 2.03, which is in accord with the Hume–Rothery criterion for the formation of the Zn-based quasicrystal phase (e/a = 2.0–2.15).

Global dependences in hexagonal tiling

Tiling is a widely used technique to solve the problems of the efficient use of multilevel memory and optimize data exchanges when developing both sequential and parallel programs. This paper investigates the problem of obtaining global dependencies, i.e. informational dependencies between tiles. The problem is solved in the context of parametrized hexagonal tiling in application to algorithms with a two-dimensional computational domain. The paper includes a formalized definition of the hexagonal tile and the criteria for dense coverage of the computational domain with hexagonal tiles. Herein, we have formulated a statement that permits to obtain all global dependencies between tiles. Formulas are constructed for the determination of sets of iterations of hexagonal tiles generating these dependencies. The sets of iterations that generate global dependencies are obtained in the form of polyhedra with an explicit expression of their boundaries.