Preston’s universal formula for avian egg shape

AmericanOrnithology.org Volume 139, 2022, pp. 1–8 https://doi.org/10.1093/ornithology/ukac028 PERSPECTIVE Preston’s universal formula for avian egg shape John D. Biggins,1,*, Robert Montgomerie,2, Jamie E. Thompson,3, and Tim R. Birkhead3,*, School of Mathematics and Statistics, The University of Sheffield, Sheffield, UK Department of Biology, Queen’s University, Kingston, Ontario, Canada 3 School of Biosciences, The University of Sheffield, Sheffield, UK *Corresponding authors: John D. Biggins, ; Tim R. Birkhead, 1 2 ABSTRACT Nearly 70 years ago, Preston published a pioneering study in which he provided formulae for the shapes of birds’ eggs. One of these formulae is universal in that it provides an almost perfect representation for all eggs, even pyriform ones, and is better than all other formulae published since. This essentially perfect representation of egg shape is obtained by estimating the parameters in Preston’s universal formula by least squares, using hundreds of measurements of the egg’s radii along its entire length. Preston’s universal formula can also be used to obtain an equation for avian egg shape that uses just 5 measurements (the length and 4 appropriately spaced diameters). The equation based solely on these 5 measurements provides an egg shape that is virtually indistinguishable from one based on hundreds of measurements. We demonstrate the usefulness of Preston’s formulations using digital photographs of eggs. Our perspective is that, despite a number of subsequent approaches, Preston’s original one has not been bettered and should be the standard for studying avian egg shape. Keywords: birds, egg geometry, egg shape LAY SUMMARY • The shapes of birds’ eggs vary considerably, from the archetypal ovoid shape of the hen’s egg, through those that are almost spherical, to eggs that are pointed at one or both ends. • In the 1950s, the engineer and amateur ornithologist Frank W. Preston published methods for measuring and quantifying avian egg shape. Unfortunately, his equations proved rather inaccessible to ornithologists and have largely been ignored. • As a result, there has been a succession of papers claiming to have solved the problem of quantifying avian egg shape, including some with titles that imply that their approach is better than anything previously published, but without testing the validity of their claims. • Here, we show that all of these papers were trying to solve a problem that Preston had resolved long ago. All methods of estimating egg shape published since the 1950s are all, to varying extents, less accurate than Preston’s original formulation. Preston provides the most perfect formula for “the most perfect thing,” a bird’s egg. Fórmula universal de Preston para la forma de huevo aviar RESUMEN Hace casi setenta años, Preston publicó un estudio pionero en el que proporcionaba fórmulas para las formas de los huevos de las aves. Una de estas fórmulas es universal porque proporciona una representación casi perfecta de todos los huevos, incluso los piriformes, y es mejor que todas las demás fórmulas publicadas desde entonces. Esta representación esencialmente perfecta de la forma del huevo se obtiene estimando los parámetros en la fórmula universal de Preston mediante mínimos cuadrados, usando cientos de medidas de los radios del huevo a lo largo de toda su longitud. La fórmula universal de Preston también puede ser usada para obtener una ecuación para la forma del huevo aviar que utiliza solo cinco medidas (la longitud y cuatro diámetros espaciados adecuadamente). La ecuación basada únicamente en estas 5 medidas proporciona una forma de huevo que es prácticamente indistinguible de una basada en cientos de medidas. Demostramos la utilidad de las formulaciones de Preston utilizando fotografías digitales de huevos. Nuestra perspectiva es que, a pesar de una serie de enfoques posteriores, el enfoque original de Preston no ha sido mejorado y debería ser el estándar para estudiar la forma del huevo aviar. Palabras clave: aves, forma del huevo, geometría del huevo Copyright © American Ornithological Society 2022. All rights reserved. For permissions, e-mail: . Submission Date: February 1, 2022; Editorial Acceptance Date: June 1, 2022; Published: June 23, 2022 2 Preston’s universal formula for avian egg shape INTRODUCTION Ornithology 139:1–8 © 2022 American Ornithological Society do they compare the effectiveness of their proposals with Preston’s. We speculate that some of the neglect of Preston’s work on egg shape is because it was too opaque for non-mathematicians. Whatever the reason, the neglect has resulted in much effort devoted to “solving” a problem that Preston (1953) had already solved more fully and more effectively, as we demonstrate in this paper. One of the obstacles to the widespread adoption of Preston’s (1953) equations was probably that his formulation used trigonometric functions. Todd and Smart (1984) rewrote Preston’s equations for egg shape in a more accessible form (see Supplementary Material, section A) that nonetheless gives shapes equivalent to Preston’s. It is Todd and Smart’s (1984) re-expression that is now commonly used (Barta and Székely 1997, Johnson et al. 2001, Mónus and Barta 2005, Frantsevich 2010, Bán et al. 2011, Biggins et al. 2018). Parameter estimation for Todd and Smart’s (1984) formulation is more straightforward than for Preston’s (1953). Specifically, having obtained measurements of egg radii (or diameters) at several locations along the egg’s length, the estimation for Todd and Smart’s (1984) formulation can be done using a general linear model—multiple regression with no intercept (see Supplementary Material, section B)—though this fact was not mentioned by Todd and Smart (1984) and seems not to be widely recognized. Preston (1953) noted that just 4 egg diameters, together with the egg length, were sufficient to estimate the 4 parameters in his Eq. (6), but he did not derive an explicit formula based on this. The fact that estimation of the parameters for Todd and Smart’s (1984) re-expression is a general linear model allows us to derive such an equation for egg shape (see Supplementary Material, section C) using the measurements illustrated in Figure 1. No such equation has been available up to now. We call the resulting formula for egg shape, which is just Preston’s Eq. (6) with parameter estimates derived from 4 specific data points, the “four-diameter equation.” The narrow aims of this paper are to demonstrate that (1) the various subsequent formulae for egg shape proposed since Preston (1953) are less effective than Preston’s universal formula (Preston 1953, his Equation 6); (2) Preston’s universal formula provides an excellent representation of shape for the entire range of avian eggs, including pyriform ones; and (3) the four-diameter equation introduced here, based on the measurements illustrated in Figure 1, is effective at describing the shapes of all birds’ egg (...truncated)