Patrones multimodales estables con simetría pentámerael caso de los Crinoideos Rhodocriníticos

Abstract

The origin of the conspicuous pentamerism oí echinoderms remains controversial. Curiously, the organization oí these living fiveíold structures has not been characterized from a mathematical point of view in any case, at any level. Starting from an amplitude equation of grandient form, triad resonant interactions determine the stability diagram oí fivefold patterns generated by five plane waves in circular regions. Each plane wave can be developed as an iníinite series oí 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 crinoids. The biological samples are compared to hydrodynamical experiments under the same general conditions of pattern selection. The multimodal structure of plate arrangement shows that many eigenstates coexist for a single set of boundary conditions which is the signature oí a highly nonlinear, and hence, highly nontrivial pattern. The present mathematical tool could be analogously useful to study echinoid skeletons.