  We present a dynamical systems approach for geometric matchings in an
ensemble of polytopes along rigid-body motions. Each polytope can be
characterized by a vertex set and edge or faces determined by vertices, and
polygons and simplexes correspond to a polytope. For a geometric matching, we
propose a system of dynamical system for the evolution of centroids and
rotations of polytopes to match the vertices under rigid-body motions which can
be decomposed as a composition of translation and rotations. Our proposed
dynamical system acts on the product space $({\mathbb R}^d \times SO(d))^N$.
The evolution of centroids can be described by the coupled linear second-order
dynamical system with diffusive linear couplings, whereas rotations for the
matching of vertices are described by the Lohe matrix model on $SO(d)^N$. In
particular, the Lohe matrix model has been derived from some set of physical
principles compared to previous works in which the Lohe matrix model were
employed as a system dynamics. This is a contrasted difference between earlier
works on the Lohe matrix model which has been adopted a priori for an aggregate
modeling of matrices. We also provide an analytical result leading to the
complete shape matchings for an ensemble of congruent polytopes, and several
numerical examples to illustrate analytical results visually.
