At a quantum critical point, the behavior of a many-body system can become largely independent of its microscopic details and instead be determined by an underlying symmetry. The one-dimensional Ising chain is a canonical example. When the system is perturbed at criticality by a magnetic field along the ordering direction, theory predicts eight particle-like excitations with mass ratios fixed by the exceptional Lie algebra $E_8$, a symmetry that has its roots in quantum field theory and high-energy physics, where bound states of particles are commonly referred to as mesons. Evidence for this $E_8$ spectrum has been reported in the insulator ${\text{CoNb}}_2{\mathrm{O}}_6$. The chains in that material are not isolated, however: coupling between them is expected to bind their excitations into meson-like states, replacing the $E_8$ spectrum with a different set of states associated with the affine algebra ${\mathcal{D}}_8^{(1)}$. This reorganization has not previously been observed directly. Here, we realize both cases in a single device using individual rubidium atoms held in focused laser beams and excited to strongly interacting states. The traps can be positioned arbitrarily, allowing us to create a closed ring of atoms, an isolated chain, or two parallel rows forming a weakly coupled ladder. We prepare each system in a simple state and let it evolve, then use spectroscopy of the resulting magnetization dynamics to extract the system’s excitation spectrum. For the ring, the measured frequencies agree with the predicted $E_8$ mass ratios. For the ladder, we observe the emergence of bound, meson-like states and signatures of the ${\mathcal{D}}_8^{(1)}$ spectrum, providing a direct view of the binding induced by interchain coupling. These results show how reconfigurable atom arrays can be used to study how emergent symmetries and their associated excitation spectra change with the geometry of a quantum system.