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Formulation and solution of the classical seashell problem

I.—Seashell geometry

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Il Nuovo Cimento D

Summary

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

Riassunto

Malgrado un’ampia e dotta letteratura che risale ai tempi classici, la geometria delle conchiglie ha resistito fino ad ora ad analisi teoriche rigorose, quindi gli scienziati che si cimentano in questo campo hanno adottato un metodo empirico senza direttiva per quanto riguarda la classificazione. L’abbondanza della recente letteratura paleontologica dimostra l’importanza di questo problema per gli scienziati di questo campo, ma non riflette in alcun modo i corrispondenti progressi concettuali o teorici rispetto al pensiero del diciannovesimo secolo che venne cosí abilmente riassunto da Sir D’Arcy Wentworth Thompson nel 1917. Tuttavia, in questo lavoro fondamentale per la nuova scienza emergente di conchigliologia teorica, l’unificazione delle considerazioni teoriche permette una rigorosa formulazione e una completa soluzione del problema della geometria biologica delle conchiglie. L’avvolgimento delle conchiglieintorno all’asse di simmetria si deduce dai primi princípi usando considerazioni sull’energia associata alla crescita per aumento di dimensioni. Questo lavoro mostra che le aperture delle conchiglie che sono incurvate (di tipo «cowrie»), allargate verso l’esterno (di tipo «strombe») o anche rivoltate all’indietro (di tipo «opistostomoideo») sono solamente casi speciali di uno spettro piú ampio di traiettorie di crescita efficienti d’energia «permesse» (spirali tensili, elastiche a molla d’orologio), molte delle quali vennero estesamente usate dagli ammoniti del Cretaceo. Considerazioni d’energia dettano anche la crescita della conchiglialungo l’asse di simmetria, cosí le spirali delle conchiglie marine possono essere comprese nei termini di certe speciali figure di rivoluzione (conoidi elastici di Möbius), essendo i gusci meglio conosciuti delle conchiglie coeloconoidali e cyrtoconoidali soltanto due casi speciali che derivano da una intera classe di geometria topologicamente possibili, efficienti di energia e biologicamente osservate. I «fili» e i «conoidi» di questo lavoro sono istruttive semplificazioni concettuali sufficienti a questo scopo. Un secondo lavoro tratterà successivamente superfici tubolari generalizzate in tre dimensioni.

Резюме

Несмотря на большое количество литературы, до настоящего времени отсутствует строгий теоретический анализ геометрией морских раковин. Многотомность существующей палеонтологической литературы демонстрирует важность этой проблемы для прикладных ученых ине отражает соответствующих концептуальных и теоретических достижений после XIX века, которые были систематизированы Томсоном в 1917 г. Однако в этой фундаментальной работе по теоретической конхиологии были объединены теоретические рассмотрения, которые впервые позволили сформулировать и полностью решить проблему биологических геометрий морских раковин. Свертывание раковин спиралью вокруг оси симметрии можно получить из первых принципов, используя энергетические рассуждения, связанные с дифференциальным ростом. В настоящей статье показывается, что те отверстия раковин, которые являются искривленными, в форме раструба, или даже повернутыми назад, представляют просто частные случаи более широкого спектра «разрешенных» траекторий роста. Энергетические рассуждения ткаже диктуют рост раковин вдоль оси симметрии, тонкое острие морских раковин можно понять в терминах некоторых специальных фигур вращения. «Провода» и «коноиды» настоящей статьи представляют поучительные концептуальные упрощения, достаточные для наших целей. Вторая статья будет посвящена трубчатым поверхностям в трех измерениях.

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Illert, C. Formulation and solution of the classical seashell problem. Il Nuovo Cimento D 9, 791–814 (1987). https://doi.org/10.1007/BF02453750

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  • DOI: https://doi.org/10.1007/BF02453750

PACS. 87.10

PACS. 46.30.Cn

PACS. 02.30.Wd

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