  Tessellations of $R^3$ that use convex polyhedral cells to fill the space can
be extremely complicated, especially if they are not facet-to-facet, that is,
if the facets of a cell do not necessarily coincide with the facets of that
cell's neighbours. In a recent paper (Weiss and Cowan, Adv.Appl.Prob. 2011), we
have developed a theory which covers these complicated cases, at least with
respect to their combinatorial topology. The theory required seven parameters,
three of which suffice for facet-to-facet cases; the remaining four parameters
are needed for the awkward adjacency concepts that arise in the general case.
This current paper establishes constraints that apply to these seven parameters
and so defines a permissible region within their seven-dimensional space, a
region which we discover is not bounded. Our constraints in the relatively
simple facet-to-facet case are also new.
