  We propose a measure based upon the fundamental theoretical concept in
algorithmic information theory that provides a natural approach to the problem
of evaluating $n$-dimensional complexity by using an $n$-dimensional
deterministic Turing machine. The technique is interesting because it provides
a natural algorithmic process for symmetry breaking generating complex
$n$-dimensional structures from perfectly symmetric and fully deterministic
computational rules producing a distribution of patterns as described by
algorithmic probability. Algorithmic probability also elegantly connects the
frequency of occurrence of a pattern with its algorithmic complexity, hence
effectively providing estimations to the complexity of the generated patterns.
Experiments to validate estimations of algorithmic complexity based on these
concepts are presented, showing that the measure is stable in the face of some
changes in computational formalism and that results are in agreement with the
results obtained using lossless compression algorithms when both methods
overlap in their range of applicability. We then use the output frequency of
the set of 2-dimensional Turing machines to classify the algorithmic complexity
of the space-time evolutions of Elementary Cellular Automata.
