  A bulge surface, on a time reflection-symmetric Cauchy slice of a holographic
spacetime, is a non-minimal extremal surface that occurs between two locally
minimal surfaces homologous to a given boundary region. According to the
python's lunch conjecture of Brown et al., the bulge's area controls the
complexity of bulk reconstruction, in the sense of the amount of post-selection
that needs to be overcome for the reconstruction of the entanglement wedge
beyond the outermost extremal surface. We study the geometry of bulges in a
variety of classical spacetimes, and discover a number of surprising features
that distinguish them from more familiar extremal surfaces such as
Ryu-Takayanagi surfaces: they spontaneously break spatial isometries, both
continuous and discrete; they are sensitive to the choice of boundary infrared
regulator; they can self-intersect; and they probe entanglement shadows,
orbifold singularities, and compact spaces such as the sphere in AdS$_p\times
S^q$. These features imply, according to the python's lunch conjecture, novel
qualitative differences between complexity and entanglement in the holographic
context. We also find, surprisingly, that extended black brane interiors have a
non-extensive complexity; similarly, for multi-boundary wormhole states, the
complexity pleateaus after a certain number of boundaries have been included.
