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Background  The globally abundant coccolithophore,  Emiliania huxleyi , plays an important ecological role in oceanic carbon biogeochemistry by forming a cellular covering of plate-like CaCO 3  crystals (coccoliths) and fixing CO 2 . It is unknown how the cells arrange different-sized coccoliths to maintain full coverage, as the cell surface area of the cell changes during daily cycle.    Methods  We used Euler’s polyhedron formula and CaGe simulation software, validated with the geometries of coccoliths, to analyze and simulate the coccolith topology of the coccosphere and to explore the arrangement mechanisms.    Results  There were only small variations in the geometries of coccoliths, even when the cells were cultured under variable light conditions. Because of geometric limits, small coccoliths tended to interlock with fewer and larger coccoliths, and vice versa. Consequently, to sustain a full coverage on the surface of cell, each coccolith was arranged to interlock with four to six others, which in turn led to each coccosphere contains at least six coccoliths.    Conclusion  The number of coccoliths per coccosphere must keep pace with changes on the cell surface area as a result of photosynthesis, respiration and cell division. This study is an example of natural selection following Euler’s polyhedral formula, in response to the challenge of maintaining a CaCO 3  covering on coccolithophore cells as cell size changes.

Coccolith arrangement follows Eulerian mathematics in the coccolithophoreEmiliania huxleyi