  A numerical semigroup is a cofinite subset of $\mathbb Z_{\ge 0}$ containing
$0$ and closed under addition. Each numerical semigroup $S$ with smallest
positive element $m$ corresponds to an integer point in the Kunz cone $\mathcal
C_m \subseteq \mathbb R^{m-1}$, and the face of $\mathcal C_m$ containing that
integer point determines certain algebraic properties of $S$. In this paper, we
introduce the Kunz fan, a pure, polyhedral cone complex comprised of a faithful
projection of certain faces of $\mathcal C_m$. We characterize several aspects
of the Kunz fan in terms of the combinatorics of Kunz nilsemigroups, which are
known to index the faces of $\mathcal C_m$, and our results culminate in a
method of "walking" the face lattice of the Kunz cone in a manner analogous to
that of a Gr\"obner walk. We apply our results in several contexts, including a
wealth of computational data obtained from the aforementioned "walks" and a
proof of a recent conjecture concerning which numerical semigroups achieve the
highest minimal presentation cardinality when one fixes the smallest positive
element and the number of generators.
