That we can construct an abstract, purely geometrical theory of morphogenesis, independent of the substrate of forms and the nature of the forces that create them, might seem difficult to believe, especially to the seasoned experimentalist used to working with living matter and always struggling with an elusive reality. This idea is not new and can be found almost explicitly in the classical book of D'Arcy Thompson, On Growth and Form, but the theories of this innovator were too far in advance of their time to be recognized; moreover, they were expressed in a geometrically naive way and lacked the mathematical justification that has only been found in the recent advances in topology and differential analysis.
This general point of view raises the following obvious question: if, according to our basic hypothesis, the only stable singularities of all morphogenesis are determined solely by the dimension of the ambient space, why do not all phenomena of our three-dimensional world have the same morphology? Why do clouds and mountains not have the same shape, and why is the form of crystals different from that of living beings? To this I reply that the model attempts only to classify local accidents of morphogenesis, which we will call elementary catastrophes, whereas the global macroscopic appearance, the form in the usual sense of the word, is the result of the accumulation of many of these local accidents. The statistic of these local accidents and the correlations governing their appearance in the course of a given process are determined by the topological structure of their internal dynamic, but the integration of all these accidents into a global structure would require, if we wanted to pursue the application of the model, a consideration of catastrophes on spaces of many more dimensions than the normal three. It is the topological richness of the internal dynamics that finally explains the boundless diversity of the external world and perhaps even the distinction between life and inert matter.