  Black holes are real astrophysical objects, but their interiors are hidden
and can only be "observed" through mathematics. The structure of rotating black
holes is typically illustrated with the help of special coordinates. But any
such coordinate choice necessarily results in a distorted view, just as the
choice of projection distorts a map of the Earth. The truest way to depict the
properties of a black hole is through quantities that are coordinate-invariant.
We compute and plot all the independent curvature invariants of rotating,
charged black holes for the first time, revealing a landscape that is much more
beautiful and complex than usually thought.
