  We investigate the qualitative features of binary black hole shadows using
the model of two extremally charged black holes in static equilibrium (a
Majumdar--Papapetrou solution). Our perspective is that binary spacetimes are
natural exemplars of chaotic scattering, because they admit more than one
fundamental null orbit, and thus an uncountably-infinite set of perpetual null
orbits which generate scattering singularities in initial data. Inspired by the
three-disc model, we develop an appropriate symbolic dynamics to describe
planar null geodesics on the double black hole spacetime. We show that a
one-dimensional (1D) black hole shadow may constructed through an iterative
procedure akin to the construction of the Cantor set; thus the 1D shadow is
self-similar. Next, we study non-planar rays, to understand how angular
momentum affects the existence and properties of the fundamental null orbits.
Taking slices through 2D shadows, we observe three types of 1D shadow: regular,
Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs
where outer fundamental orbits are forbidden by angular momentum. The highly
chaotic part is associated with an unexpected feature: stable and bounded null
orbits, which exist around two black holes of equal mass $M$ separated by $a_1
< a < \sqrt{2} a_1$, where $a_1 = 4M/\sqrt{27}$. To show how this possibility
arises, we define a certain potential function and classify its stationary
points. We conjecture that the highly chaotic parts of the 2D shadow possess
the Wada property. Finally, we consider the possibility of following null
geodesics through event horizons, and chaos in the maximally-extended
spacetime.
