The Epistemology of Biomimetics: The Role of Models and of Morphogenetic Principles

Form follows function, but it does not follow from function. Form is not derivable from the latter. To realize a desired technical function, a form must first be found that is able to realize it at all. Secondly, the question arises as to whether an envisaged form realizes the function in an appropriate way. Functions are multiply realizable—various different forms can bear the very same function. One needs to find a form of a technical artifact that realizes an envisaged function sufficiently efficient, robust, or whatever criteria might be imposed. This paper scrutinizes biomimetics as one way to find a good solution to the realization problem. Drawing on an approach from the philosophy of simulations, it reconstructs the biomimetic relation as being mediated by a theoretical model. It is shown that the robustness of the functioning system is usually reached in different ways in biological and in technological systems, which explains differences in morphogenetic mechanisms or principles found in these fields. This reconstruction helps to understand problems with robustness in synthetic biology that occur when technical design principles are implemented in a biological system. The mimetic relation between the biological and the technical realm is found to be asymmetric.

Exotic Nonleaves with Infinitely Many Ends

We show that any simply connected topological closed $4$-manifold punctured along any compact, totally disconnected tame subset $\Lambda $ admits a continuum of smoothings, which are not diffeomorphic to any leaf of a $C^{1,0}$ codimension one foliation on a compact manifold. This includes the remarkable case of $S^4$ punctured along a tame Cantor set. This is the lowest reasonable regularity for this realization problem. These results come from a new criterion for nonleaves in $C^{1,0}$ regularity. We also include a new criterion for nonleaves in the $C^2$-category. Some of our smooth nonleaves are “exotic”, that is, homeomorphic but not diffeomorphic to leaves of codimension one foliations on a compact manifold, being the 1st examples in this class.

The Realization Problem for Discrete Morse Functions on Trees

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if they induce the same persistence diagram. We compare this notion of equivalence to other notions of equivalent discrete Morse functions. Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree. This is a version of the “realization problem” of the persistence map. We conclude with an example illustrating our construction.