Formulation and solution of the classical seashell problem

摘要: Despite an extensive scholarly literature dating back to classical times, seashell geometries have hitherto resisted rigorous theoretical analysis, leaving applied scientists adopt a directionless empirical approach toward classification. The voluminousness of recent palaeontological demonstrates the importance this problem scientists, but in no way reflects corresponding conceptual or advances beyond XIX century thinking which was so ably summarized by Sir D’Arcy Wentworh Thompson 1917. However, foundation paper for newly emerging science conchology, unifying considerations first time, permits formulation and complete solution biological shell geometries. Shell coilingabout axis symmetry can be deduced from principles using energy associated with incremental growth. present shows that those apertures are incurved («cowrielike»), outflared («stromblike») even backturned («Opisthostomoidal») merely special cases much broader spectrum «allowable» energy-efficient growth trajectories (tensile elastic clockspring spirals), many were widely used Cretaceous ammonites. Energy also dictate growthalong symmetry, thus spires understood terms certain figures revolution (Mobius conoids), better-known coeloconoidal cyrtoconoidal being only two arising whole class topologically possible, efficient biologically observed «wires» «conoids» instructive simplifications sufficient purposes. A second will later deal generalized tubular surfaces three dimensions.