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Packings of regular convex polygons (n-gons) that are sufficiently dense have been studied extensively in the context of modeling physical and biological systems as well as discrete and computational geometry. Formerly most of the results were on densest lattice or double-lattice configurations. Here we extend these results to all 2-dimensional crystallographic symmetry groups (plane groups) by restricting the configuration space of the general packing problem of congruent copies of a compact subset of the 2-dimensional Euclidean space to particular isomorphism classes of the discrete group of isometries. We formulate the plane group packing problem as a nonlinear constrained optimization problem. By the means of the Entropic Trust Region Packing Algorithm that solves this problem approximately, we examine some known as well as unknown densest packings of n-gons in all 17 plane groups and state conjectures about common symmetries of the densest plane group packings of all n-gons.

It has been demonstrated that the computational capabilities of echo state networks are maximized when the recurrent layer is close to the border between a stable and an unstable dynamics regime, the so called edge of stability, or criticality. The maximization of performance is computationally useful, leading to minimal prediction error or maximal memory capacity, and has been shown to lead to maximization of information-theoretic measures, such as transfer entropy and active information storage in case of some datasets. In this paper, we take a closer look at these measures, using Kraskov-Grassberger-Stogbauer estimator with optimized parameters. We experiment with four datasets differing in the data complexity, and discover interesting differences, compared to the previous work, such as more complex behavior of the information-theoretic measures. We also investigate the effect of reservoir orthogonalization, that has been shown earlier to maximize memory capacity, on the prediction accuracy and the above mentioned measures.