YET ANOTHER MATHEMATICAL MODEL OF EGGS: TWO-PARAMETRIC BRANDT'S SHAPES

The explanation of the perfect forms in nature is a subject which has attracted a great attention since the beginning of our civilization. E.g., despite the longstanding interest in the shapes of the eggs, the available parametric descriptions in the modern literature are given only via purely empirical formulas without any clear relationships with their measurable parameters. Here we are exploring another geometrical model suggested by G. Brandt long time ago but which has not received any proper treatment. Actually, it is based on fourth order plane algebraic curve for which we have derived several parameterizations. The most interesting of them is the uniformization of the curve via the Jacobian elliptic functions. Some comparisons with the experimental results were made but this subject deserves further investigation.

Automated summary

The explanation of perfect forms in nature has been of great interest throughout history. However, current parametric descriptions of shapes, such as eggs, lack clear relationships with measurable parameters and rely solely on empirical formulas. In this study, we explore a geometric model proposed by G. Brandt, based on a fourth-order plane algebraic curve, and derive several parameterizations. The most fascinating one is the uniformization of the curve using Jacobian elliptic functions. Comparisons with experimental results have been made, but further investigation is needed.