Egg and math: introducing a universal formula for egg shape

Abstract

The egg, as one of the most traditional food products, has long attracted the attention of mathematicians, engineers, and biologists from an analytical point of view. As a main parameter in oomorphology, the shape of a bird's egg has, to date, escaped a universally applicable mathematical formulation. Analysis of all egg shapes can be done using four geometric figures: sphere, ellipsoid, ovoid, and pyriform (conical or pear-shaped). The first three have a clear mathematical definition, each derived from the expression of the previous, but a formula for the pyriform profile has yet to be derived. To rectify this, we introduce an additional function into the ovoid formula. The subsequent mathematical model fits a completely novel geometric shape that can be characterized as the last stage in the evolution of the sphere-ellipsoid-Hügelschäffer's ovoid transformation, and it is applicable to any egg geometry. The required measurements are the egg length, maximum breadth, and diameter at the terminus from the pointed end. This mathematical analysis and description represents the sought-for universal formula and is a significant step in understanding not only the egg shape itself, but also how and why it evolved, thus making widespread biological and technological applications theoretically possible.

Mathematical progression of avian egg shape with associated area and volume determinations

“…Whereas calculation of spheres and ellipsoids is given in geometric reference books, to obtain adequate dependences for ovoids, whose shape obeys Hügelschäffer's model, we carried out a series of theoretical and experimental studies 18,19,21 that resulted in the following formulae: Thus, the problem was narrowed down to obtaining the calculated dependences V pyr and S pyr for pyriform eggs. As we showed earlier, 3 the boundary shape for such eggs is the combination of the paraboloid at the pointy end and the ovoid of Hügelschäffer's model at the blunt end (Fig. 1).…”

“…It should be noted that for all types of eggs, the blunt end shape corresponds to Hügelschäffer's model. 3 That is, keeping the same principle of assigning subscript indices we have adopted, the volume and surface area at the blunt end of pyriform eggs are denoted by V pyr(b) and S pyr(b) , which will be identical to V con(b) (Eq. 13) and S con(b) (Eq.…”

“…Earlier, as a defining indicator of the contour of eggs of any shape, we justified the use of the diameter D L/4 of the egg at the point corresponding to 1 4 of its length, that is, taken at a distance of L/4 from the pointy end of the egg. 3 At the same time, D L/4 for an ovoid egg corresponding to Hügelschäffer's model will be determined as twice the value of y from formula (11) after substituting the value x = L/4. D L/4 for a conical egg will be similarly defined as twice the value of y from formula (6) after the same appropriate substitution.…”

Hugelschaffer egg curve and surface

“…Analogous to the previous example, a Hügelschäffer egg-shaped curve model can be generated according to the profiles of egg-shaped pipe sewers obtained through composition of circular arcs with the shape index B/L = 2/3 or B/L = 3/4 as given in [47]. In recent years, the application of Hügelschäffer models, as well as other mathematical models for describing the shape of eggs -ovoid forms (curves and surfaces) in the poultry industry and food engineering is on the rise, as evident from papers [2], [13], [14], [16], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28].…”

“…The reason for the use of the Hügelschäffer egg-shaped curve model is its presentation as a non-destructive oomorphological model when calculating the area of a planar curve A and the volume V and the surface area S of a 3D egg surface [23], [27], [28]. The authors Narushin et al used 2D digital images of egg contours to obtain concrete values for the parameters L = 2a, B = 2b and w, [22], [23].…”