  Frieze groups are discrete subgroups of the full group of isometries of a
flat strip. We investigate here the dynamics of specific architected materials
generated by acting with a frieze group on a collection of self-coupling seed
resonators. We demonstrate that, under unrestricted reconfigurations of the
internal structures of the seed resonators, the dynamical matrices of the
materials generate the full self-adjoint sector of the stabilized group
$C^\ast$-algebra of the frieze group. As a consequence, in applications where
the positions, orientations and internal structures of the seed resonators are
adiabatically modified, the spectral bands of the dynamical matrices carry a
complete set of topological invariants that are fully accounted by the K-theory
of the mentioned algebra. By resolving the generators of the K-theory, we
produce the model dynamical matrices that carry the elementary topological
charges, which we implement with systems of plate resonators to showcase
several applications in spectral engineering. The paper is written in an
expository style.
