Tiling mechanisms of the Drosophila compound eye through geometrical tessellation.

Tiling patterns are observed in many biological structures. The compound eye is an interesting example of tiling and is often constructed by hexagonal arrays of ommatidia, the optical unit of the compound eye. Hexagonal tiling may be common due to mechanical restrictions such as structural robustness, minimal boundary length, and space-filling efficiency. However, some insects exhibit tetragonal facets. Some aquatic crustaceans, such as shrimp and lobsters, have evolved with tetragonal facets. Mantis shrimp is an insightful example as its compound eye has a tetragonal midband region sandwiched between hexagonal hemispheres. This casts doubt on the naive explanation that hexagonal tiles recur in nature because of their mechanical stability. Similarly, tetragonal tiling patterns are also observed in some Drosophila small-eye mutants, whereas the wild-type eyes are hexagonal, suggesting that the ommatidial tiling is not simply explained by such mechanical restrictions. If so, how are the hexagonal and tetragonal patterns controlled during development? Here, we demonstrate that geometrical tessellation determines 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. We propose that 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.