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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.

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