Measurement of the neutral axis in avian eggshells reveals which species conform to the golden ratio

A novel approach to egg and math: Improved geometrical standardization of any avian egg profile

Developing a geometric formulation of any biological object has a number of justifications and applications. Recently, we developed a universal geometric figure for describing a bird's egg in any of the possible basic shapes: spherical, ellipsoidal, ovoid, and pyriform. The formulation proved widely applicable but had a number of drawbacks, including a very obvious join between two parts of the egg. To correct this, we developed the Main Axiom of the universal mathematical formula. This essentially involved making the ordinate of the extremum of the function correspond to half the maximum egg breadth (B), and the abscissa to the reciprocal of the parameter w that reflects the shift of the vertical axis to its coincidence with B. This, in turn, helped us develop a new, simplified mathematical model without a nonbiological join. Experimental verification was performed to confirm the adequacy of the new geometric figure. It accurately described actual avian eggs of various shapes more closely than our previous model. To the best of our knowledge, our new, simplified equation can be applied as a standard for any bird egg that exists in nature. As a rather simple equation, it can be used in a broad range of applications.

Egg-inspired engineering in the design of thin-walled shelled vessels: a theoretical approach for shell strength

A novel subdiscipline of bionics is emerging in the form of ‘egg-inspired engineering’ through the use of egg-shaped ovoids as thin-walled tanks and building structures. Hügelschäffer’s and Narushin’s models of egg geometry are highly applicable within this proposed subdiscipline. Here we conducted a comparative analysis between the two models with respect to some of the most important egg parameters. These included contents volume, shell volume, and the location of the neutral axis along the shell thickness. As a first step, theoretical studies using the Narushin’s model were carried out due to the lack (or limited amount) of data on the geometric relationships of parameters and available calculation formulae. Considering experimental data accumulated in the engineering and construction industries, we postulate a hypothesis that there is a correlation between location of the neutral axis and the strength of the walls in the egg-shaped structure. We suggest that the use of Narushin’s model is preferable to Hügelschäffer’s model for designing thin-walled shelled vessels and egg-shaped building structures. This is due to its relative simplicity (because of the requirement for only two initial parameters in the basic equation), optimal geometry in terms of material costs per unit of internal capacity, and effective prerequisites for shell strength characteristics.