Universal Von Neumann Neighborhood Cellular Automata on Penrose Tilings

Cellular Automata on the Penrose tilings (PTCA) have come to draw a lot of attention recently. We constructed previously a 6-state universal semi-totalistic Moore neighbor-hood Kite and Dart PTCA by asynchronous circuit simulation. Tsukamoto et, al constructed a 4-state universal Rhomb PTCA by synchronous circuit simulation. In this paper we investigated the rotation-symmetric von Neumann neighborhood case. We show a 5-state universal Kite and Dart PTCA in the case of asynchronous circuit simulation and two 3-state PTCAs capable of simulating a set of universal logical elements in the case of synchronous circuit simulation.