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1. 3. Recalling some results already presented
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The logistic model, just mentioned as a simple example, is useful in a lot of different contexts but, as recalled above, it is particularly useful to describe the dynamic growth of biological populations; variants of it are also used to model several aspects of the growth of single organisms [8]. Thus it is not surprising that one of the first classical accounts on the strange behaviours of the model is due to a biologist: Robert May [9].It is well-known that Klee maintained a deep interest in organic growth throughout his entire artistic career. Thus, it was quite obvious to me trying to apply the model to the analysis of both Klee’s painting (by means of computer simulations) and writings. The encouraging results are presented in a paper [10]. Interestingly, Klee guessed that quite different behaviours (that we termed above as asymptotic, oscillating and chaotic) can be seen as different particular cases of the same dynamics, and this can be easily shown for both paintings and writings of Klee’s.In the same paper I also discussed the use of other biomathematical models, such as Turing’s reaction-diffusion mechanism; by means of cellular automata, I applied Turing’s model to simulate several modules of the late Klee. Interestingly, I also indicated in Klee’s writings significant and surprising contact points between Turing’s mechanism and the dynamic behaviour of systems, as conceived by Klee himself. Once again from a single model we obtain very different pictures, especially depending on the procedures used to complete the image, but also depending on the different possible values of parameters, which we know imply different kinds of attractor for the system; in Turing’s mechanism periodic attractors give rise to labyrinthine striped or spotted patterns, whereas strange attractors determines patterns with chaotic waves. The most part of Klee’s late style modules can be traced back to the latter chaotic behaviour of the system. In a second paper (already recalled in note [7]), I discussed some chaotic structures, which Klee obtained by means of iterative procedures inside the circle: we are talking about iterative and fully deterministic methods which however led Klee to those typical, highly irregular and whirling sketches and drawings, which we often may properly name tangles. In the same paper I linked the chaos theme as conceived in Klee to other themes, relevant in complexity sciences, equally investigated by Klee (such as recursion, circular feedback, fractals, instabilities, self-organisation and so on). Other artists in the first half of the century shared a deep interest in the same tightly connected themes; in the cited paper I discussed several parallelisms between Klee, Escher and Duchamp. |
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