  The Laplace operator encodes the behavior of physical systems at vastly
different scales, describing heat flow, fluids, as well as electric,
gravitational, and quantum fields. A key input for the Laplace equation is the
curvature of space. Here we discuss and experimentally demonstrate that the
spectral ordering of Laplacian eigenstates for hyperbolic (negatively curved)
and flat two-dimensional spaces has a universally different structure. We use a
lattice regularization of hyperbolic space in an electric-circuit network to
measure the eigenstates of a "hyperbolic drum", and in a time-resolved
experiment we verify signal propagation along the curved geodesics. Our
experiments showcase both a versatile platform to emulate hyperbolic lattices
in tabletop experiments, and a set of methods to verify the effective
hyperbolic metric in this and other platforms. The presented techniques can be
utilized to explore novel aspects of both classical and quantum dynamics in
negatively curved spaces, and to realise the emerging models of topological
hyperbolic matter.
