  Fingerprint distances, which measure the similarity of atomic environments,
are commonly calculated from atomic environment fingerprint vectors. In this
work we present the simplex method which can perform the inverse operation,
i.e. calculating fingerprint vectors from fingerprint distances. The
fingerprint vectors found in this way point to the corners of a simplex. For a
large data set of fingerprints, we can find a particular largest volume
simplex, whose dimension gives the effective dimension of the fingerprint
vector space. We show that the corners of this simplex correspond to landmark
environments that can by used in a fully automatic way to analyse structures.
In this way we can for instance detect atoms in grain boundaries or on edges of
carbon flakes without any human input about the expected environment. By
projecting fingerprints on the largest volume simplex we can also obtain
fingerprint vectors that are considerably shorter than the original ones but
whose information content is not significantly reduced.
