  Networks of interacting gyroscopes have proven to be versatile structures for
understanding and harnessing finite-frequency topological excitations. Spinning
components give rise to band gaps and topologically protected wave transport
along the system's boundaries, whether the gyroscopes are arranged in a lattice
or in an amorphous configuration. Here, we examine the irrelevance of periodic
order for generating topological gaps. Starting from the symplectic dynamics of
our model metamaterial, we present a general method for predicting whether a
gap exists and for approximating the Chern number using only local features of
a network, bypassing the costly diagonalization of the system's dynamical
matrix. We then study how strong disorder interacts with band topology in
gyroscopic metamaterials and find that amorphous gyroscopic Chern insulators
exhibit similar critical behavior to periodic lattices. Our experiments and
simulations additionally reveal a topological Anderson insulation transition,
wherein disorder drives a trivial phase into a topological one.
