Discretization-Dependent Dissolution of (Dis)Continuous Gliders: Non-Platonic Self-Organization in Complex Systems

Abstract: Many simulated complex systems that support persistent self-organizing patterns, i.e. gliders, have a 'state-plus-update' paradigm. This approach can be found in computational models of physics, continuous and neural cellular automata, residual connections in neural networks, and optimization methods like stochastic gradient descent. If the update is the output of a differential equation and modulated by a step size parameter, we have the familiar and general Euler method. Generally, a smaller step size is expected to yield more accurate results, at the expense of more computations to arrive at a desired end point. I examine multiple systems supporting gliders that fit into the Euler method framework, including multiple approaches to continuous cellular automata and the Gray-Scott artificial chemistry system. Each of these systems yield one or more glider pattern-rule pairs that persist under specific, and sometimes quite coarse, discretization conditions, but become unstable at nominally more accurate, finer simulation conditions. These patterns (in combination with the systems they persist in) are clearly not approximations approaching an abstract ideal as discretization tends to zero, but exist on their own, somewhat baffling, terms that include the systematic errors of particular discretization regimes. I refer to these gliders as 'non-Platonic'. Code for replicating or expanding on this work has been made available at https://github.com/RiveSunder/DiscoGliders.