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