Complex systems are often modeled as Boolean networks in attempts to capture their logical structure and reveal its dynamical consequences. Approximating the dynamics of continuous variables by discrete values and Boolean logic gates may, however, in
troduce dynamical possibilities that are not accessible to the original system. We show that large random networks of variables coupled through continuous transfer functions often fail to exhibit the complex dynamics of corresponding Boolean models in the disordered (chaotic) regime, even when each individual function appears to be a good candidate for Boolean idealization. A suitably modified Boolean theory explains the behavior of systems in which information does not propagate faithfully down certain chains of nodes. Model networks incorporating calculated or directly measured transfer functions reported in the literature on transcriptional regulation of genes are described by the modified theory.
This paper addresses the question of whether a single tile with nearest neighbor matching rules can force a tiling in which the tiles fall into a large number of isohedral classes. A single tile is exhibited that can fill the Euclidean plane only wit
h a tiling that contains k distinct isohedral sets of tiles, where k can be made arbitrarily large. It is shown that the construction cannot work for a simply connected 2D tile with matching rules for adjacent tiles enforced by shape alone. It is also shown that any of the following modifications allows the construction to work: (1) coloring the edges of the tiling and imposing rules on which colors can touch; (2) allowing the tile to be multiply connected; (3) requiring maximum density rather than space-filling; (4) allowing the tile to have a thickness in the third dimension.
