Finite automata working bitwise on the local coordinates of points in the plane are constructed and shown to lead to self-affine surfaces ('patchwork quilts') under general circumstances. 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 .
Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all bitwise operations on a finite alphabet which are not governed by chance. It is shown that these models give rise to a roughness exponent H that shapes the resulting spatial patterns , larger values of the exponent leading to coarser surfaces .
R
Finite automata working bitwise on the local coordinates of points in the plane are constructed and shown to lead to self-affine
Fractal
R
under general circumstances. We prove
are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all bitwise operations on a finite alphabet which are not governed by chance. It is shown
A
H
R
: Larger
, larger
R
lead
leading
D
. 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