Universality in two dimensions

Turing, in his immortal 1936 paper, observed that '[human] computing is normally done by writing... symbols on [two-dimensional] paper', but noted that use of a second dimension 'is always avoidable' and that 'the two-dimensional character of paper is no essential of computation'. We propose to promote two-dimensional models of computation and exploit the naturalness of two-dimensional representations of data. In particular, programs for a two-dimensional Turing machine can be recorded most naturally on its own two-dimensional input-output grid in such a transparent fashion that schoolchildren would have no difficulty comprehending their behaviour. This two-dimensional rendering allows, furthermore, for a most perspicacious rendering of Turing's universal machine.