Broadcasting Automata and Patterns on Z^2

The Broadcasting Automata model draws inspiration from a variety of sources such as Ad-Hoc radio networks, cellular automata, neighbourhood sequences and nature, employing many of the same pattern forming methods that can be seen in the superposition of waves and resonance. Algorithms for broadcasting automata model are in the same vain as those encountered in distributed algorithms using a simple notion of waves, messages passed from automata to automata throughout the topology, to construct computations. The waves generated by activating processes in a digital environment can be used for designing a variety of wave algorithms. In this chapter we aim to study the geometrical shapes of informational waves on integer grid generated in broadcasting automata model as well as their potential use for metric approximation in a discrete space. An exploration of the ability to vary the broadcasting radius of each node leads to results of categorisations of digital discs, their form, composition, encodings and generation. Results pertaining to the nodal patterns generated by arbitrary transmission radii on the plane are explored with a connection to broadcasting sequences and approximation of discrete metrics of which results are given for the approximation of astroids, a previously unachievable concave metric, through a novel application of the aggregation of waves via a number of explored functions. Thomas Nickson The University of Edinburgh, Kennedy Tower, Royal Edinburgh Hospital, Edinburgh EH10 5HF e-mail: [email protected] Igor Potapov University of Liverpool, Ashton Street, Ashton Building, Liverpool, L69 3BX, e-mail: potapov@