  We consider the numerical evolution of black hole initial data sets,
consisting of single black holes distorted by strong gravitational waves, with
a full 3D, nonlinear evolution code. These data sets mimic the late stages of
coalescing black holes. We compare various aspects of the evolution of
axisymmetric initial data sets, obtained with this 3D code, to results obtained
from a well established axisymmetric code. In both codes we examine and compare
the behavior of metric functions, apparent horizon properties, and waveforms,
and show that these dynamic black holes can be accurately evolved in 3D. In
particular we show that with present computational resources and techniques,
the process of excitation and ringdown of the black hole can be evolved, and
one can now extract accurately the gravitational waves emitted from the 3D
Cartesian metric functions, even when they carry away only a small fraction
($<< 1%$) of the rest mass energy of the system. Waveforms for both the
$\ell=2$ and the much more difficult $\ell=4$ and $\ell=6$ modes are computed
and compared with axisymmetric calculations. In addition to exploring the
physics of distorted black hole data sets, and showing the extent to which the
waves can be accurately extracted, these results also provide important
testbeds for all fully nonlinear numerical codes designed to evolve black hole
spacetimes in 3D, whether they use singularity avoiding slicings, apparent
horizon boundary conditions, or other evolution methods.
