An Algorithmic Approach to Tilings of Hyperbolic Spaces: Universality Results

Abstract

In this paper, our results on algorithmic analysis of tiling in hyperbolic spaces are discussed. We overview results and developments obtained by the approach, focusing on the construction of universal cellular automata in hyperbolic spaces with a minimal number of cell states.

Group theory and p‐adic valued models of swarm behaviour

“…The same pattern with distinguishing the 2 stages in the propagation can be seen everywhere in the swarm behaviour, including a phenomenon of diffusion given by the growth of colonies of bacteria, see Ben‐Jacob and Margenstern . In Margenstern,() there was proposed a simulation of the logistic stage by cellular automata in the hyperbolic plane, where there are innitely many different tilings. One of the examples of the space for these cellular automata is given by the tiling when 3 copies of polygon can be put around a vertex to cover a neighbourhood of the vertex with no overlapping.…”

“…In all cases, the other neighbours are increasingly numbered from 2 to 7 while counterclockwise turning around the cell starting from side 1. For more details, see Margenstern . In this way, the logistic stage of the swarm propagation is simulated by a possibly infinite behaviour within a rigid limits.…”

Construction and Shape Optimization of Simplicial Meshes in d-Dimensional Space

“…This is based on translation invariance which does not hold in non-euclidean geometries. More on tessellations of hyperbolic spaces can be found in works of Coxeter [5] or [6], and Margenstern [18], [19], [20]. As Margenstern points out, these works might find their use in computational problems of theory of relativity or cosmological research, but such results had not been published before 2003 and to the best author's knowledge not even since these days.…”