Modelling Natural Patterns and Forms II: (Easter) Eggs

﻿When you start with a portrait and search for a pure form,
a clear volume, through successive eliminations, you arrive
inevitably at the egg. Likewise, starting with the egg and
following the same process in reverse, one finishes with
the portrait.

PABLO PICASSO

Easter is coming and what better time to talk about eggs!

During my recent mathematical explorations of natural shapes and forms, my attention has been catched by the shape of birds eggs. In the interesting book by J. Adams, A Mathematical walk in Nature [1], you can find a short review on the different mathematical modelling approach to describe the shape of an egg. Among them, the geometrical one by Baker [2] is revealed one of the most versatile as it can very accurately reproduce the shapes of a large variety of bird eggs [2]. More recently, the model was used to perform a systematic and comparative study of the shape of bird eggs. This study, published on Science magazine [3], a two-dimensional morphological space defined by the parameters of the Baker’s equation, has been used to show the diversity of the shape of 1400 species of birds. Combining these information with a mechanical model and phylogenetics information, the authors have shown that egg shape correlates with flight ability on broad taxonomic scales. They concluded that adaptations for flight may have been critical drivers of egg-shape variation in birds [3].

The Baker’s equation is based is based on a fundamental transformation from projective geometry. This mathematical description generates so-called ‘‘path curves’’ that remarkably resemble natural shapes such as eggs, buds and embrios. This model was originally developed by L. Edwards [4] based on previous idea of other mathematicians. The details of the derivation can be in the referenced publications, here I just talk about its application. The equation is given by the following expression

$y=T(1+x)^{\frac{\lambda}{1+\lambda}}(1-x)^{\frac{1}{1+\lambda}}$

where $T$ and $\lambda$ are two parameters. In the paper by Stoddard et al. [3], the asymmetry ($A=\lambda-1$) and ellipticity ($E=\frac{1}{T}-1$) parameters are used instead to define the charateristic of the analyzed eggs. For $A=0$ and $E=0$, the egg has a spherical shape, as in the figure below.

An example of European bird that make eggs with low value of A and E is the European blue bee eater.

If the value of the parameter $E$ increase the egg shape become more elliptic. For example, with A=0 and E=1, we get the egg in the Figure 2.

An example of bird egg with low value of A and larger value of E is, for example, the one from the Great bustard bird (see figure below).

If the value of the parameter $A$ increase the egg shape become more elliptic. For example, with A=1 and E=0, we get the egg in the Figure 3.

An example of bird egg with large value of A and larger value of E is, for example, the one from the Least sandpiper (see figure below).

It is possible to calculate the volume and the surface of the solid of the revolution generated by the Beker’s equation. If L is the average length (L) of the egg main axis then the volume of revolution is

$V=\frac{\pi L^3}{8}\int_{-1}^1 y^2 dx$

and the surface are

$S=\frac{\pi L^2}{2}\int_{-1}^1 y \sqrt {1+\frac{dy}{dx} } dx.$

The egg models in this blog has been generated using a tcl/tk program that I have written for the occasion. The program has a database of more than 300 european birds. The value of A and E in the database has been taken from the supplementary data of Stoddard et al paper [3] and the bird common names from IOC word list of bird names. The current version fo the program can calculate the volume and the surface of the egg by integrating the revolution solid using the Simpson 1/3 rule.

What about choccolate egg shapes? Well, for the moment shall we just enjoy them!

BUONA PASQUA – HAPPY EASTER – FROHE OSTERN

and stay tuned …

REFERENCES

1. J. A. Adams. A Mathematical Nature Walk. Princeton University Press. 2011.
2. D. E. Baker. A Geometric Method for Determining Shape of Bird Eggs. The Auk 119(4):1179–1186, 2002.
3. M. C. Stoddard et al. Avian egg shape: Form, function, and evolution.Science 356, 1249–1254 (2017).
4. L. Edwards. The Vortex of Life: Nature’s Patterns in Space and Time. Floris Books, 2nd Revised edition edition (4 May 2006).