  Coordination geometries describe how the neighbours of a central particle are
arranged around it. Such geometries can be thought to lie in an abstract
topological space; a model of this space could provide a mathematical basis for
understanding physical transformations in crystals, liquids, and glasses. With
this motivation, the present work proposes a metric model of the space of
three-dimensional coordination geometries. This model is conceived through the
generalisation of a local orientational order parameter and seems to be
consistent with geometric intuition. It appears to suggest a taxonomy of
coordination geometries with five main classes, each with a distinct character.
A quantitative notion of orientational typicality is introduced and its
interplay with orientational order is found to evidence a statistical
regularity with respect to point symmetry. By the assertion of axioms on the
topology of the space herein modelled, the range of structures that are
possible to resolve with the order parameter in molecular dynamics simulations
is greatly increased.
