Comparison of two polar equations in describing the geometries of domestic pigeon (Columba livia domestica) eggs

Two-dimensional (2D) egg-shape equations are potent mathematical tools, facilitating the description of avian egg geometries in their applied mathematical modelling and poultry science implementations. They aid in the precise quantification of avian egg sizes, including traits such as volume (V) and surface area (S). Despite their potential, however, polar coordinate egg-shape equations have received relatively little attention for practical applications in oomorphology. This may be attributed to their complex model structure and the absence of explicit geometric interpretations for the equation parameters. In the present study, two distinct polar equations, namely the Carter-Morley Jones equation (CMJE) and simplified Gielis equation (SGE), were used to fit the profile geometries of 415 domestic pigeon (Columba livia domestica) eggs based on nonlinear least squares regression methods. The adequacy of goodness-of-fit for each nonlinear egg-shape equation was evaluated through the adjusted root-mean-square error (RMSEadj), while relative curvature measures of nonlinearity were utilized to assess the nonlinear behavior of equations. All of the RMSEadj values of the two polar equations were lower than 0.05, which demonstrated the validity of CMJE and SGE in depicting the shapes of C. livia egg profiles. Moreover, the two egg-shape equations showed good nonlinear behavior across all 415 C. livia eggs. Wilcoxon signed rank tests relative to RMSEadj values between CMJE and SGE revealed that CMJE displayed inferior fits to empirical data when compared to SGE. CMJE, however, had a better linear approximation performance than SGE at the global level. At the individual parameter level, all of the parameters of CMJE or SGE exhibited good close-to-linear behavior. This study provides an instrumental mathematical tool for the practical application of polar egg-shape equations, such as non-destructively estimating V and S of avian eggs. Additionally, it offers valuable insights into assessing nonlinear regression models for accurately describing the geometries of 2D egg profiles

Conquering Mount Improbable

Abstract: Our scientific and technological worldviews are largely dominated by the concepts of entropy and complexity. Originating in 19th-century thermodynamics, the concept of entropy merged with information in the last century, leading to definitions of entropy and complexity by Kolmogorov, Shannon and others. In its simplest form, this worldview is an application of the normal rules of arithmetic. In this worldview, when tossing a coin, a million heads or tails in a row is theoretically possible, but impossible in practice and in real life. On this basis, the impossible (in the binary case, the outermost entries of Pascal's triangle xn and yn for large values of n) can be safely neglected, and one can concentrate fully on what is common and what conforms to the law of large numbers, in fields ranging from physics to sociology and everything in between. However, in recent decades it has been shown that what is most improbable tends to be the rule in nature. Indeed, if one combines the outermost entries xn and yn with the normal rules of arithmetic, either addition or multiplication, one obtains Lam\ue9 curves and power laws respectively. In this article, some of these correspondences are highlighted, leading to a double conclusion. First, Gabriel Lam\ue9's geometric footprint in mathematics and the sciences is enormous. Second, conic sections are at the core once more. Whereas mathematics so far has been exclusively the language of patterns in the sciences, the door is opened for mathematics to also become the language of the individual. The probabilistic worldview and Lam\ue9's footprint can be seen as dual methods. In this context, it is to be expected that the notions of information, complexity, simplicity and redundancy benefit from this different viewpoint