In his Bakerian Lecture, Bernal (1964) discussed those ideas of restricted irregularity which are physically realized in random packings of equal hard spheres, with particular reference to the structure of simple liquids. He stressed the need for a science of 'statistical geometry', and took the first steps himself by proposing possible ways of describing such arrays. In this paper, these and other associated ideas are briefly described, and extended by deriving an equivalent set of polyhedral subunits essentially inverse to the packing in real space. Examination of two independent high density arrays demonstrates the reproducibility of certain metrical and topological properties of these polyhedra, and their correlations over larger elements of volume. As a result, several possible 'descriptive parameters' are proposed. Although these essentially 'numerical' characteristics facilitate sensitive structural descriptions of any assembly of micro- and macroscopic subunits, we are still unable to characterize an irregular array in formal mathematical terms. Such a formulation of statistical geometry could be a powerful tool for tackling important problems in many branches of science and engineering.