  Adjacent dyadic systems are pivotal in analysis and related fields to study
continuous objects via collections of dyadic ones. In our prior work (joint
with Jiang, Olson and Wei) we describe precise necessary and sufficient
conditions for two dyadic systems on the real line to be adjacent. Here we
extend this work to all dimensions, which turns out to have many surprising
difficulties due to the fact that $d+1$, not $2^d$, grids is the optimal number
in an adjacent dyadic system in $\mathbb{R}^d$. As a byproduct, we show that a
collection of $d+1$ dyadic systems in $\mathbb{R}^d$ is adjacent if and only if
the projection of any two of them onto any coordinate axis are adjacent on
$\mathbb{R}$. The underlying geometric structures that arise in this higher
dimensional generalization are interesting objects themselves, ripe for future
study; these lead us to a compact, geometric description of our main result. We
describe these structures, along with what adjacent dyadic (and $n$-adic, for
any $n$) systems look like, from a variety of contexts, relating them to
previous work, as well as illustrating a specific example.
