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In his 1952 paper "The chemical basis of morphogenesis", Alan M. Turingpresented a model for the formation of skin patterns. While it took severaldecades, the model has been validated by finding corresponding naturalphenomena, e.g. in the skin pattern formation of zebrafish. More surprising,seemingly unrelated pattern formations can also be studied via the model, likee.g. the formation of plant patches around termite hills. In 1984, David A.Young proposed a discretization of Turing's model, reducing it to anactivator/inhibitor process on a discrete domain. From this model, the conceptof three-dimensional Turing-like patterns was derived.  In this paper, we consider this generalization to pattern-formation inthree-dimensional space. We are particularly interested in classifying thedifferent arising sub-structures of the patterns. By providing examples for thedifferent structures, we prove a conjecture regarding these structures withinthe setup of three-dimensional Turing-like pattern. Furthermore, we investigate- guided by visual experiments - how these sub-structures are distributed inthe parameter space of the discrete model. We found two-fold versions of zero-and one-dimensional sub-structures as well as two-dimensional sub-structuresand use our experimental findings to formulate several conjectures forthree-dimensional Turing-like patterns and higher-dimensional cases.

Experimental visually-guided investigation of sub-structures in three-dimensional Turing-like patterns