  We explore the dependence on spatial dimension of the viability of the
neutrino heating mechanism of core-collapse supernova explosions. We find that
the tendency to explode is a monotonically increasing function of dimension,
with 3D requiring $\sim$40$-$50\% lower driving neutrino luminosity than 1D and
$\sim$15$-$25\% lower driving neutrino luminosity than 2D. Moreover, we find
that the delay to explosion for a given neutrino luminosity is always shorter
in 3D than 2D, sometimes by many hundreds of milliseconds. The magnitude of
this dimensional effect is much larger than the purported magnitude of a
variety of other effects, such as nuclear burning, inelastic scattering, or
general relativity, which are sometimes invoked to bridge the gap between the
current ambiguous and uncertain theoretical situation and the fact of robust
supernova explosions. Since real supernovae occur in three dimensions, our
finding may be an important step towards unraveling one of the most problematic
puzzles in stellar astrophysics. In addition, even though in 3D we do see
pre-explosion instabilities and blast asymmetries, unlike the situation in 2D,
we do not see an obvious axially-symmetric dipolar shock oscillation. Rather,
the free energy available to power instabilites seems to be shared by more and
more degrees of freedom as the dimension increases. Hence, the strong dipolar
axisymmetry seen in 2D and previously identified as a fundamental
characteristic of the shock hydrodynamics may not survive in 3D as a prominent
feature.
