  Many-body descriptors are widely used to represent atomic environments in the
construction of machine learned interatomic potentials and more broadly for
fitting, classification and embedding tasks on atomic structures. It was
generally believed that 3-body descriptors uniquely specify the environment of
an atom, up to a rotation and permutation of like atoms. We produce several
counterexamples to this belief, with the consequence that any classifier,
regression or embedding model for atom-centred properties that uses 3 (or
4)-body features will incorrectly give identical results for different
configurations. Writing global properties (such as total energies) as a sum of
many atom-centred contributions mitigates, but does not eliminate, the impact
of this fundamental deficiency -- explaining the success of current
"machine-learning" force fields. We anticipate the issues that will arise as
the desired accuracy increases, and suggest potential solutions.
