_{ 3 } crystals (coccoliths) and fixing CO_{ 2 }.It is unknown how the cells arrange different-size of coccoliths to maintainfull coverage, as the cell surface area of the cell changes during daily cycle.

_{ 3 }covering on coccolithophore cells as cell sizechanges.

The manuscript was restructured, and more simulation results (see Figs. 4-5)

were added to more clearly explain our points. The following text simply explains our points: As the basic component of the coccosphere, coccoliths

are produced with specific geometry. Thus, we propose that the formation of coccospheres

must follow basic mathematical principles or constraints.

coccoliths on a 2D plane. We found that, because of geometric limits, small

coccoliths tended to interlock with fewer and larger coccoliths, and vice versa.

Thus, each coccolith interlocks with four to six coccoliths. Second, we used Euler’s

formula and the software CaGe to analyze the coccolith topology on the

coccosphere. The simulation results on the 2D plane and 3D polyhedrals matched

very well. Therefore, the coccosphere contains at least six coccoliths to

sustain full coverage. In summary, this study validated the geometries of the coccolith

and demonstrated that the noted coccolith arrangement pattern is the only mathematical

solution to form coccospheres. We also discussed the geometric limit on the

effective coverage area of coccolith, and how coccolith number per coccosphere adjusted

to changing cell surface area during photosynthesis, respiration, and cell

division. Finally, we applied our methods to analyze the coccolith topology of

a specific coccolithophore,

The polygon compositions of polyhedra (6-22 faces) were predicted based on Euler’s formula, and further examined using the software CaGe to obtain isomer numbers. The face numbers are a list of consecutive positive integers. The isomer number of polyhedral coccosphere was calculated using the software CaGe. The numbers in red color indicate where no polyhedron exist with predicted polygon compositions.

The authors declare that they have no competing interests.