  In this work, we study the cellular decomposition of $S$ induced by a filling
pair of curves $v$ and $w$, $Dec_{v,w}(S) = S - (v \cup w)$, and its connection
to the distance function $d(v,w)$ in the curve graph of a closed orientable
surface $S$ of genus $g$. Efficient geodesics were introduced by the first
author in joint work with Margalit and Menasco in 2016, giving an algorithm
that begins with a pair of non-separating filling curves that determine
vertices $(v,w)$ in the curve graph of a closed orientable surface $S$ and
computing from them a finite set of efficient geodesics. We extend the tools of
efficient geodesics to study the relationship between distance $d(v,w)$,
intersection number $i(v,w)$, and $Dec_{v,w}(S)$. The main result is the
development and analysis of particular configurations of rectangles in
$Dec_{v,w}(S)$ called spirals. We are able to show that, in some special cases,
the efficient geodesic algorithm can be used to build an algorithm that reduces
$i(v,w)$ while preserving $d(v,w)$. At the end of the paper, we note a
connection of our work to the notion of extending geodesics.
