  How can we manipulate the topological connectivity of a three-dimensional
prismatic assembly to control the number of internal degrees of freedom and the
number of connected components in it? To answer this question in a
deterministic setting, we use ideas from elementary number theory to provide a
hierarchical deterministic protocol for the control of rigidity and
connectivity. We then show that is possible to also use a stochastic protocol
to achieve the same results via a percolation transition. Together, these
approaches provide scale-independent algorithms for the cutting or gluing of
three-dimensional prismatic assemblies to control their overall connectivity
and rigidity.
