Arnold weak resonance equation as the model of Greek ornamental design

Abstract

We propose and study a mathematical model that qualitatively reproduces several ancient ornamental designs than one can see in historical museums of Crete and Athens. The designs contain several rings that circumscribe a fixed number of centers or spirals, specific to each design. The model is based on a complex differential equation of weak resonance (Arnold 1977). We analyze the role of the model parameters in giving rise to different peculiarities of the repeated designs, in particular, the dynamical indeterminacy. The model allows tracing design changes under parameter variation, as well as to construct some new ornamental designs. We discuss how observed ornamental design may reflect some philosophical ideas of ancient inhabitants of Greece.