Langton's flies

Abstract

Langton's ant is an automaton defined over a two-dimensional grid. Its movement is governed by the environment in a very simple way: it turns to the left over white vertices and to the right over black vertices. This definition only applies on a two-dimensional space. We look for generalizations of this automaton to n-dimensional lattices. Remembering the different ways through which the ant was originally defined, we consider two approaches: the first comes from physics (lattice gas) and the second from artificial life (virtual ants). Two generalizations are proposed defining two families of dynamical systems. From the physics point of view, the ant is seen as a particle and hence it has no internal state other than its velocity. From the artificial life point of view, the ant is viewed as an insect, and it has an orientation in space which we represent by an orthogonal basis. This constitutes the ant's internal state. This formulation allows us to define the ant's behaviour without drawing upon any information relative to the global system of external coordinates. Each model yields different sets of rules with distinctive behaviours. We characterize all the possible rules satisfying some basic restrictions. We found that many rules produce trajectories which are restricted to a diagonal plane and are equivalent to a version of Langton's ant over a two-dimensional grid, squared or hexagonal. In the particle model, only two of them use the whole space, and it is shown that such rules do not admit periodical trajectories. This result reinforces a previous one reported by Leonid Bunimovich, who states that ‘the skeleton of any bounded trajectory cannot contain any three-dimensional polyhedron’.