  In recent papers, we and colleagues have introduced a way to visualize the
full vacuum Riemann curvature tensor using frame-drag vortex lines and their
vorticities, and tidal tendex lines and their tendicities. We have also
introduced the concepts of horizon vortexes and tendexes and 3-D vortexes and
tendexes (regions where vorticities or tendicities are large). Using these
concepts, we discover a number of previously unknown features of quasinormal
modes of Schwarzschild and Kerr black holes. These modes can be classified by
mode indexes (n,l,m), and parity, which can be electric [(-1)^l] or magnetic
[(-1)^(l+1)]. Among our discoveries are these: (i) There is a near duality
between modes of the same (n,l,m): a duality in which the tendex and vortex
structures of electric-parity modes are interchanged with the vortex and tendex
structures (respectively) of magnetic-parity modes. (ii) This near duality is
perfect for the modes' complex eigenfrequencies (which are well known to be
identical) and perfect on the horizon; it is slightly broken in the equatorial
plane of a non-spinning hole, and the breaking becomes greater out of the
equatorial plane, and greater as the hole is spun up; but even out of the plane
for fast-spinning holes, the duality is surprisingly good. (iii)
Electric-parity modes can be regarded as generated by 3-D tendexes that stick
radially out of the horizon. As these "longitudinal," near-zone tendexes rotate
or oscillate, they generate longitudinal-transverse near-zone vortexes and
tendexes, and outgoing and ingoing gravitational waves. The ingoing waves act
back on the longitudinal tendexes, driving them to slide off the horizon, which
results in decay of the mode's strength. (iv) By duality, magnetic-parity modes
are driven in this same manner by longitudinal, near-zone vortexes that stick
out of the horizon. [Abstract abridged.]
