Most computability results on subshifts are proved on $\mathbb Z^2$ by embedding arbitrary computations of Turing machines, which classically consists in drawing the space-time diagram of the Turing machine (one dimension being used for space, one dimension being used for time) ; and then, said results generalize to higher dimensions by reducing to the two-dimensional case.

These reductions between the $2$ and the $d$-dimensional case have left a gap in my understanding of how we compute with subshifts: how do you embed arbitrary computations in a $d$-dimensional tiling? For example, if you draw the space-time diagram of a Turing machine on $\mathbb Z^3$, do you use one dimension for space, and two for time? This appears to suggest a tradeoff between time and space when dealing with $d$-dimensional computations.

In this talk, we show that there is, in fact, no tradeoff. By considering "mesh-connected multicomputers" as a computation model, one can implement the sorting of an array of size $[0,n]^{d-1}$ into a tiling of the $[0,n]^d$ cube. From this, we can embed $O(n^{d-1})$ steps of arbitrary word-RAM computations into a tiling of the $[0,n]^d$ cube.