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

TL;DR

This study explores three-dimensional Turing-like patterns through experiments and classification, confirming conjectures about their sub-structures and distribution in parameter space, advancing understanding of complex pattern formation.

Contribution

It classifies sub-structures in 3D Turing-like patterns, proves a related conjecture, and investigates their parameter space distribution through visual experiments.

Findings

Identified two-fold zero- and one-dimensional sub-structures

Discovered two-dimensional sub-structures in patterns

Formulated conjectures for higher-dimensional Turing-like patterns

Abstract

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.