Three into two doesn»t go: two-dimensional models of bird eggs, snail shells and plant roots

Abstract

Three published two-dimensional analyses of three-dimensional structures are reinvestigated. (1) In their 1997 paper, Barta & Székely sought shapes of bird eggs that maximized the size of eggs that packed under a circular brood patch. Inappropriately they measured egg size by cross-sectional area; maximizing volume implies different optima. Also their genetic algorithm mislocated their optima, probably because of too much mutation. (2) In this journal in 1985, Heath considered a section of an idealized snail shell; altering the overlap of adjacent whorls alters the ratio of shell material to volume enclosed. Heath located an overlap that minimized perimeter/area of the cross-section, which is different from minimizing surface-area/volume. I derive the surface area of a logarithmic helicospiral; some formulae used previously are slightly incorrect. Heath's alteration of overlap changed shell volume and aperture area; a more meaningful reanalysis keeps these characters constant. (3) In 1987, Fitter used a two-dimensional model of plant roots to show that topology, growth rate, and nutrient diffusion rate affect exploitation efficiency (area of nutrient depletion zone/volume of root). Recalculation and reanalysis show that only part of the effect of topology depends on overlap of depletion zones. I compare Fitter's model to coplanar root systems with three-dimensional depletion zones and then to fully three-dimensional branching. Finally I discuss further examples in which two-dimensional models could mislead.