Before revision:  We prove that these models give rise to a roughness exponent  that shapes the resulting spatial patterns : Larger values of the exponent lead to coarser surfaces . We suggest that finite automata provide the mathematical link between the concept of positional information of modern theoretical biology and the emergence of fractal self-affine surfaces ubiquitously found in nature .
After revision:  We prove that these models give rise to a roughness exponent that shapes the resulting spatial patterns : Larger values of the exponent lead to coarser surfaces .