Hyperincursive methods for generating fractals in automata related to diffusion and wave equations

This paper provides modelling tools for automata design in the field of information and formal systems. The concept and methods of incursion and hyperincursion are firstly applied to the Fractal Machine, a hyperincursive cellular automaton with sequential computations with exclusive OR, where time plays a central role. Simulations will show the generation of fractal patterns. The computation is incursive, for inclusive recursion, in the sense that an automaton is computed at the future time t + 1 as a function of its neighbour automata at the present and/or past time steps but also at the future time t + 1. The hyperincursion is an incursion when several values can be generated at each time step. External incursive inputs cannot be transformed to recursion. (This is really an example of the Final Cause of Aristotle.) But internal incursive inputs defined at the future time can be transformed to recursive inputs by self-reference. A particular case of self-reference with the Fractal Machine shows a non deterministic hyperincursive field. Interference of particles in the Fractal Machine gives rise to fractal patterns which follow Huygens Principle of Secondary Sources. The superimposition of states is similar to Deutsch's quantum computing principle. This is also related to digital cellular automata obtained from diffusion and wave finite difference equations. Zuse proposed to represent physical systems by a computing space based on such a digitalisation of differential equations. I show that the digital wave equation exhibits waves by digital particles with interference effects and uncertainty.