Patrones multimodales estables con simetría pentámera: el caso de los Crinoideos Rhodocrinítidos

Abstract

The origin of the conspicuous pentamerism of echinoderms remains controversial. Curiously, the organization of diese living fivefoid structures has not been characterized from a matitematical point of view in any case, at any level. Starting from an amplirude equation of grandient form, triad rcsonarn inieractions determine Pie stability diagram of fivefoid pattems generated by five plane waves in circular regions. Each plane wave can be developed as an infinite series of Bessel modes. However, boundary constraints reduce the number of modes to a discrete set. This general approach is used here to study the skeleton plate arrangement of bowl or globose cup crínoids. The biological samples are compared lo hydrodynamical experimcnts under the sarne general conditions of pattern selection. The muhimodal structure of plate anangemenr shows that many elgenstates coexist for a single set of boundary conditions which is tite signature of a highty nonlinear, and hence, highíy nontrivial patlern.The present mathematical tool could be analogously useful lo study echinoid skeletons.