The use of spherical harmonics in the molecular sciences is widespread. They have been employed with success in, for instance, the crystallographic fast rotation function, small-angle scattering particle reconstruction, molecular surface visualisation, protein–protein docking, active site analysis and protein function prediction. An extension of the spherical harmonic expansion method is presented here that enables regions (bodies) rather than contours (surfaces) to be described and which lends itself favourably to the construction of rotationally invariant shape descriptors. This method introduces a radial term that extends the spherical harmonics to 3D polynomials. These polynomials maintain the advantages of the spherical harmonics (orthonormality, completeness, uniqueness and fast computation) but correct the drawbacks (contour based shape description and star-shape objects) and give rise to powerful invariant descriptors. We provide proof-of-principle examples illustrating the potential of this method for accurate object representation, an analysis of the descriptor classification power, and comparisons to other methods.