  For any constant $d$ and parameter $\varepsilon > 0$, we show the existence
of (roughly) $1/\varepsilon^d$ orderings on the unit cube $[0,1)^d$, such that
any two points $p,q\in [0,1)^d$ that are close together under the Euclidean
metric are "close together" in one of these linear orderings in the following
sense: the only points that could lie between $p$ and $q$ in the ordering are
points with Euclidean distance at most $\varepsilon\| p - q \|$ from $p$ or
$q$. These orderings are extensions of the $\mathcal{Z}$-order, and they can be
efficiently computed. Functionally, the orderings can be thought of as a
replacement to quadtrees and related structures (like well-separated pair
decompositions). We use such orderings to obtain surprisingly simple algorithms
for a number of basic problems in low-dimensional computational geometry,
including (i) dynamic approximate bichromatic closest pair, (ii) dynamic
spanners, (iii) dynamic approximate minimum spanning trees, (iv) static and
dynamic fault-tolerant spanners, and (v) approximate nearest neighbor search.
