Chapter 5: Two Dimensions and Beyond

Section 3: Turing Machines

History [of 2D Turing machines]

At a formal level 2D Turing machines have been studied since at least the 1950s. And on several occasions systems equivalent to specific simple 2D Turing machines have also been constructed. In fact, much as for cellular automata, more explicit experiments have been done on 2D Turing machines than 1D ones. A tradition of early robotics going back to the 1940s—and leading for example to the Logo computer language—involved studying idealizations of mobile turtles. And in 1971 Michael Paterson and John Conway constructed what they described as an idealization of a prehistoric worm, which was essentially a 2D Turing machine in which the state of the head records the direction of the motion taken at each step. Michael Beeler in 1973 used a computer at MIT to investigate all 1296 possible worms with rules of the simplest type on a hexagonal grid, and he found several with fairly complex behavior. But this discovery does not appear to have been followed up, and systems equivalent to simple 2D Turing machines were reinvented again, largely independently, several times in the mid-1980s: by Christopher Langton in 1985 under the name "vants"; by Rudy Rucker in 1987 under the name "turmites"; and by Allen Brady in 1987 under the name "turning machines". The specific 4-state rule

{s_, c_} With[{sp = s (2c - 1) }, {sp, 1 - c, {Re[sp], Im[sp]}}]

has been called Langton's ant, and various studies of it were done in the 1990s.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]