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On growth and form and geometry. I. (English) Zbl 1349.53032
The authors discuss an approach to the biological theory of growth and form, using a geometrical model for the planar natural growth of D’Arcy Thompson and its thereafter formulated natural \(nD\) generalization and the geometrical forms which these models do characterize: the logarithmic spirals or the equiangular curves of Descartes amongst the curves in Euclidean planes \(\mathbb E^2\), and the constant ratio or equiangular submanifolds of Bang-Yen Chen amongst \(nD\)-submanifolds \(M^n\) in Euclidean \((n+m)D\)-spaces \(\mathbb E^{n+m}\). These ideas are explained by the example of the surface of any shell, and some others. The extended basic principle of growth however is the same for all dimensions \(n\) and for all co-dimensions \(m\), and besides in biology, may be applied in other fields of science too. The authors consider this problem for pseudo-Euclidean ambient spaces and illustrate this one by examples.

53B25 Local submanifolds
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry