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

Abstract Avian eggs represent a striking evolutionary adaptation for which shell thickness is crucial. An understudied eggshell property includes the neutral axis, a line that is drawn through any bent structure and whose precise location is characterized by the k‐factor. Previous studies have established that, for chicken eggs, mean k corresponds to the golden ratio (Φ = 1.618, or 0.618 in its reciprocal form). We hypothesized whether such an arrangement of the neutral axis conforms to the eggshell of any bird or only to eggshells with a certain set of geometric parameters. Implementing a suite of innovative methodological approaches, we investigated variations in k of 435 avian species, exploring which correspond to Φ. We found that mean k is highly variable among birds and does not always conform to Φ, being much lower in spherical and ellipsoid eggs and higher in pyriform eggs. While 21 species had k values within 0.618 ± 0.02 (including four falcon species) and the Falconinae subfamily (six species) revealed a mean of 0.618, it is predominantly domesticated species (chicken, ducks, and geese) that lay eggs whose neutral axis corresponds to the golden ratio. Thus, the study of the mathematical secrets of the eggshell related to the golden ratio of its neutral axis suggests its species‐specific signatures in birds.


INTRODUCTION
In the evolution of terrestrial vertebrates, one of the most striking adaptations is the formation, laying, brooding, and hatching of calcified avian eggs. Essentially, the bird's egg needs to have a series of characteristics for which its shape, parameters, and thickness are crucial. It needs to be large enough to incubate an embryo but small enough, and this beam to a profile of specific interest. Here, the measurement of l becomes more complex as the outer part of the beam is somewhat stretched, and the inner part shrunken. The true value of l, therefore, is somewhere in between these extremes. These problems are not new for engineers, materials scientists, builders, and so on, and the calculation of such a midline is called the neutral axis. 5 The neutral axis corresponds to an imaginary line that has the same length before the beam was bent. 6 Moreover, its precise location is characterized by the so-called k-factor, a ratio of the neutral axis location compared to the material thickness. In other words, the formula is k = t/T, with t being the neutral axis position and T being the material thickness. 7 Crucially, the neutral axis is not always at the exact middle of T, that is, k is not always equal to 0.5. Rather, k depends on the bend radius and material thickness, 8 on the properties and composition of the material, 9 and finally on the forces applied to bend the beam. 10 In other words, the neutral axis is an imaginary line in a beam or other structure subjected to a bending action in which the fibers are neither stretched nor compressed, or where the longitudinal stress is zero.
Imagine now a scenario where no force was applied to the material, but, rather, it was already formed curved. In nature, one such example is the shell of a bird's egg, which is typically spherical, ellipsoid, ovoid, or pyriform in shape. 11 Knowledge of its various internal and external parameters has wide applications in ornithological curation, the poultry industry, and ecological studies. Rahn and Paganelli 12 stated that no studies have yet been able to obtain accurate shell density estimates using oological collections as they usually rely on only approximate calculation methods. To the best of our knowledge, this holds true over 20 years later. Such detailed calculations are also relevant for the food industry, that is, in commercial poultry farming, when it is necessary to calculate the volume of the shell and, accordingly, the contents of the egg. In a recent study, 13 we established that the nondestructive calculation of the shell volume is much more accurate if we use the exact value of the k-factor and not an approximate value of 0.5, as is conventionally accepted. Indeed, most previous studies that considered oomorphology assume the neutral axis to be at the midline, that is, k = 0.5. The purpose of this study was to challenge this notion and thereby calculate k for a range of eggs spanning the phylogenetic class Aves. In order to do this, we developed the theoretical prerequisites as outlined in File S1.
Expressing k from the obtained Equation (S1.18) (see File S1) is a laborious task, so in this context, we analyzed it using numerical methods, substituting into Equation (S1.18) various sets of parameters characteristic, say, for chicken eggs. In addition to the already clarified variety of values for L/T = [120 . . . 230], we needed to decide on the nature of changing B/L and w/L. The former is no more than the so-called shape index proposed by Romanoff and Romanoff 14 that, according to their data, varies for chicken eggs in the range of [0.65 . . . 0.82]. This was also confirmed by the data of our previous experiments. [15][16][17][18][19][20][21] The w/L ratio is not an index as commonly known as B/L, and, therefore, we could only rely on our data produced by measuring these parameters. 21 Although in another work, 3 by dividing each ratio range into subcategories, we generated 378   different combinations, thus obtaining 378 cubic equations based on Equation (S1.18). Solving each of them using Cardan's formulae, 22 we obtained different variations in k values from 0.327 to 0.959, with an average value of 0.618. It was this number, which also reflects the location of the neutral axis in the shell of chicken eggs, that made us examine it more closely. The reason is that the value of k exactly corresponded to the value of the so-called golden ratio. 23 The golden ratio (Φ = 1.618, or 0.618 in its reciprocal form) is well described in mathematics, science, architecture, and the arts and is otherwise known as the golden section, golden mean, or divine proportion.
It can be traced back to Euclid and is defined as "the ratio of a line segment cut into two pieces of different lengths such that the ratio of the whole segment to that of the longer segment is equal to the ratio of the longer segment to the shorter segment." 24 With the golden ratio in mind, we tested here the hypothesis that such a unique arrangement of the neutral axis corresponds typically the shell of any bird, the alternative being that it only applies to the shells of eggs with a certain set of geometric parameters. To do this, we implemented the methodological approaches described in the Methods section.

METHODS
The generation of all possible combinations of characteristics to encompass the entire variety of bird eggs is an unfeasible task. For this reason, we decided to rely, as a starting point, on the previously adopted model species and its eggs, that is, the chicken, which is the most widespread and useful, both in terms of variability and broad experimental applications. 25 The study of the influence of the geometric dimensions of eggs on the position of the shell's neutral axis was carried out by the method of correlation analysis. For this, we used the statistical and mathematical algorithms available in the STATIS- We also focused on examining the data available for the shells of various wild bird species. Data cards for some museum collections of avian eggs are somewhat patchy; 26 however, the most complete, both in terms of species diversity and availability of a set of oomorphological data, is likely to be the Schönwetter 27  Along with the various egg images, Schönwetter 27 also presented information about 23 eggs of the extinct elephant bird (genus Aepyornis, infraclass Palaeognathae-increasing our species count by one, to 435) that, in addition to the sizes B, L, and T that were of interest to us, included data on their volumes, V. Considering the formula we derived earlier for recalculating w from V as follows: 1 indicators for this species were also used for the analyses.
Recalculation according to Equation (1) made it possible to obtain rational w values only for a sample of 11 eggs, which was due to some inaccuracies in Schönwetter's 27 determination of actual egg volumes.
Thus, to the set of 434 images of bird eggs, the values of the parameters for the 11 Aepyornis eggs were also added. Each egg image was analyzed as follows. Measurements of geometric parameters were carried out in pixels using the Microsoft Office Picture Manager program. First, sizes B and L were measured. Thereafter, the egg radiuses were measured at points corresponding to the L/4 length, respectively, from the sharp and blunt edges, which we will denote as y L/4 (measured from a sharp edge) and y -L/4 (measured from a blunt edge). Narushin et al. 3 showed that the shape of an egg can be classified as a classical ovoid, corresponding to the Hügelschäffer model, if the radius of the egg at a point distant from the sharp edge by the value of L/4 is not lower than y L/4 calculated by the following formula: Accordingly, at y L∕4 = √ 3B∕4, the ovoid will match a sphere, and if y L/4 is less than that calculated by Equation (2), the egg falls into the pyriform (pear-shaped) category. 4 If the measured value of y L/4 is equal to the value of y -L/4 , then the shape of such an egg will correspond to an ellipsoidal one.
Narushin et al. 21 demonstrated that even using 2-D digital methods, it is quite difficult to measure the value of w, and therefore, we proposed a method for recalculating it through the measured values of y L/4 and y -L/4 : This calculation formula of Equation (3)

Classical egg-shaped ovoid
The classification of eggs according to their profile was reasonably well established. In our previous work; 3 in this study, and as mentioned in the Introduction, we used the gradation of Nishiyama, 11 who proposed dividing the shape of eggs into four main geometric groups: spherical, ellipsoidal, ovoid, and pyriform (conical). Since chicken eggs can be considered the classic representatives of eggs with an ovoid profile, 3 we chose this group for a more detailed analysis of the location of the neutral axis of their shell. Using the resulting database of 378 generated chicken egg sizes, we derived an empirical dependence for calculating k that was more convenient and simpler than the theoretical Equation (1) (see the Methods): with R 2 = 0.9998.
Also, using the generated data, we estimated the degree of influence of each of the three indices on the value of the coefficient k, as presented in the form of graphical dependencies ( Figure 1).
It seemed to us a very interesting property of ovoid shells that the indices containing only the external parameters (B, L, and w) have a much greater effect on their internal characteristic (k) than the index that only includes the direct internal parameter, the shell thickness (T).
We thus carried out the further analysis in such a way as to find an answer to the question: at what parameters of the three egg indices does the coefficient k have the value of the golden ratio, 0.618? For this purpose, we introduced the golden ratio value into Equation (4): Thus, Equation (5) In each of the above equations (i.e.,

Analysis of eggshells based on a reference
To test the above hypothesis, and also in view of the fact that, despite the wide intraspecific variability of the simulated parameters, they still remain typical for the group of chicken eggs, we proceeded to the analysis of other bird species, the egg sizes of which were obtained by measuring their images from the oological reference book by Schönwetter. 27 The results of the measurements and the corresponding calculations are presented in Table 1.
According to the data in Table 1, the mean value of k does not correspond to the value of the golden ratio (0.618), exceeding it by a factor of ∼1.5 (Table 1, Table 2. A further subdivision of birds led to more detailed information on mean k variations for eggs of all studied orders ( Figure 2A) and separately for passerine families ( Figure 2B).
Of these, only 21 species approximated the golden ratio (Table 3).
Notably, four of these were in the order Falconidae and three of the egg, when comparing the three taxonomic groups, that is, Aepyornis, passerines, and non-passerines. In particular, Aepyornis had a thicker shell than non-passerines, whereas passerines turned out to be the most thin-shelled. Since the group of passerines is most recent in evolutionary terms, 33 we attempted, for each index, to track any possible dynamics of this process. For this, using the TimeTree web tool [30][31][32] that enabled us to judge the approximate date of origination and divergence of a particular taxon, we plotted graphical dependences shown in Figure 3A-D.

TA B L E 3
List of species whose k value fell within 0.02 of the golden ratio  As we have already noted, the shape of all bird eggs can be divided into four main categories. 11 The spherical shell can be defined, theoretically, with relative ease, as we have done above in the introduction section of this work. However, among eggs of various shapes considered in our analysis, there were a few attributed to an ellipsoidal shape and a limited number of pyriform ones. We explore these below in more detail.

Ellipsoid
Eggs of this shape are relatively common in nature. How will the neutral axis behave in the shell of ellipsoidal eggs? It was not difficult to address this question, and for this, we took the value of w/L = 0 in Equation (S1.8) (see File S1), since the central vertical axis of the ellipsoids is not displaced and runs along its maximum width. Then, Equation (S1.8) can be rewritten as: Substituting into Equation (9)

Pyriform (conical) ovoid
Using a similar, already established methodology for calculating the coefficient k for passing the neutral axis in the shell of conical eggs, we applied the calculated formulae derived by us 4 to determine their volume, V con , and surface area, S con : Performing transformations similar to our theoretical studies for spherical and ovoid eggs, we produced a cubic equation for calculating k (see its detailed derivation presented in File S2):   34 Ar et al., 35 Rahn and Paganelli, 12 and Peterson et al., 36 we generated 378 different combinations, thus obtaining 378 cubic equations based on Equation (12).
Solving each of them, we obtained variations for k values characteristic of pyriform ovoids, that is, from 1.75 to 4.72, with an average value of 3.139, which was very far from the golden ratio.
Among the sample of eggs from the oological reference book, 27

DISCUSSION
The study herein presented represents the largest and most comprehensive overview of the neutral axis of bird eggshells to date. Despite among domesticated birds, such as chickens, ducks, and geese (see Introduction), we reject the hypothesis that such a figure is typical of avian eggs in general. Rather, the k value is highly variable, much lower in spherical and ellipsoid eggs and higher in pyriform eggs, varying considerably even among passerine birds. The golden ratio for the neutral axis, therefore, only applies to domesticated species, such as chickens, ducks, and geese, plus the handful of species indicated in the study for whom an approximate value of 0.618 was found (Table 3) Attempts to link the parameters of bird eggs with the golden ratio have been made many times. Both the egg and the golden ratio are important objects in architecture and art, and also have a certain touch of mystery 23,37-42 that is presumably the stimulus for a number of researchers to try and correlate them. Of course, the first thing that could be tied into this relationship is the ratio of egg length and maximum breadth, that is, the basic index B/L, called the shape index. 14 Although it is not difficult to find an egg with the ratio of these parameters equal to 0.618, the average values of this index are slightly higher.
As our studies of the eggs of wild bird species have shown, on average, it corresponds to 0.74 and differs little from this value when divided into main shape groups (Tables 1 and 2 giving them the task of drawing an egg profile in the hope of proving that the golden ratio is such an elegant shape that, in the imagination of an ordinary person, the contour ratio would be just that. However, she invariably received an average result with a clearly overestimated value. 43 In point of fact, in nature, the most common B/L ratio equal to 0.618 is found in geese (see an example in Figure 5). Perhaps not without reason that the famous Aesop's fable 44 referred to "the goose that laid the golden egg," although there are other versions of the fable that refer to a chicken rather than a goose. 45 Mytiai and Matsyura 46 also found in some species that the ratio of blunt and pointed ends radiuses can be described with the golden ratio. Petrović and Obradović 47 considered in their work many options for plotting the contours of eggs, using as a basis for such a design a combination of various parameters, the ratio of which corresponds to the golden ratio. Levine 48 and later Nagy 49 obtained egg profiles by substituting different values of n in the basic equation x n + y n = 1 so that n conforms to the value of the golden ratio raised to a certain power. Some of the profiles obtained did indeed resemble an egg, although most were far from the classic ovoids. Considering these studies together, it seems that attempts to link together the golden ratio and egg geometry are somewhat artificial. Indeed, choosing measurements of certain parameters, the ratio of which is 0.618, is quite simple, both for a variety of natural and manmade objects, by virtue of the fact that, with such a large sample size, some are bound to conform by chance alone (Table 3). We thus do not exclude the possibility that our study case is probabilistic; however, when the probability consistently repeats itself, it is reasonable to hypothesize that this may be a pattern. Such a constancy has only been noted in the case of parameters corresponding to chicken eggs, and possibly eggs of other poultry species. In this context, we can rely on the data based on measurements of duck eggs [50][51][52] and goose eggs. [53][54][55][56] In other words, only those eggs of bird species intensively domesticated by humans appear to conform, notwithstanding the 21 species found in this study (Table 3)  Given that the neutral axis golden ratio applies predominantly to species domesticated by humans over many generations, it seems reasonable to suggest that it was directed artificial selection that contributed to the production of ovoids with these "golden" geometric parameters. It is highly unlikely, of course, that poultry breeders deliberately targeted this criterion when domesticating poultry. Judging from the graph in Figure 2A, it is possible that at the stage of their evolution, the parameters of the earliest eggs corresponded to this "golden" principle. However, with each evolutionary leap, the neutral axis ran lower and lower, that is, the value of k increased steadily.
This fact was due to a relative decrease in shell thickness, that is, an increase in the L/T index ( Figure 2B). This becomes particularly apparent after the appearance of the passerine species. Similar results were obtained by Rahn and Paganelli, 12 who, like us, used data from the oological reference book. 27 They derived two different allometric relationships for calculating the shell thickness for non-Passeriformes and Passeriformes, since in the latter, with a similar egg mass, the shell thickness was always less. Perhaps, it was one of several specific features of passerines that contributed to their widespread distribution Thus, to ensure that the value of k corresponded to the golden ratio, the evolutionary process of changing w/L would have to go in the opposite direction, that is, by decreasing the value of the parameter w and, respectively, approaching the geometric shape of the egg toward the ellipsoid. However, in reality, this has not happened. Surely, the relationship of the oomorphological parameters is much deeper than the attempt to provide the location of the neutral axis of the eggshell. We can suggest that a more compact laying of eggs in the nest depends on the size w, [63][64][65] and, consequently, their optimal air and temperature exchange and, as a result, incubation success. In any event, it seems that if the golden ratio of the shell plays a particular role for bird eggs (see File S3 for further discussion), then this factor is clearly inferior to other egg quality characteristics (e.g., shape [2][3][4] ) that are no less important and relevant.
Importantly, our study is a useful finding given that the golden ratio has been asserted in numerous contexts in earlier studies, despite the fact that those assertions were not always accurate (indeed, they were frequently false). Some of them 66,67 argued the existence of the golden ratio in self-replicating systems, but this is not particularly unique in view of various other existing mathematical constants. Others 68,69 believed that it is simple to find non-golden ratios in many natural systems. Therefore, our study offers crucial information to help resolve such contradictions in the context of eggshells.

CONCLUSION
Collectively, the present study exemplifies an original attempt to look nondestructively inside the eggshell, revealing its characteristics and secrets using the appropriate mathematical apparatus only. Moreover, we have for the first time designed, described, and successfully applied a method to easily recalculate the location of the neutral axis in any geometrical object. A few novel indices have been introduced to characterize avian eggs comprehensively in relation to their evolutionary adaptation. Our findings contribute to developing a conception of how the mathematical secrets of eggshells can reveal new evolutionary signatures in birds.

ACKNOWLEDGMENTS
The skilled technical assistance of Olga M. Romanova in preparing Figure 2A is kindly appreciated.