New PNAS article by Eva Miranda: Two-dimensional billiards can simulate any Turing machine
Sep 21, 2026
A single ball bouncing across a billiard table can compute anything a computer can compute. This is the conclusion of an article by Professor Eva Miranda (UPC–CRM) and Isaac Ramos, Eva Miranda’s master’s student at ETH Zürich, recently published in the journal Proceedings of the National Academy of Sciences (PNAS).
Billiards are one of the most classical models of deterministic motion: a particle moves freely and bounces specularly off rigid walls. The authors demonstrate that, even in two dimensions, billiard trajectories can simulate arbitrary Turing machines. The result resolves an open problem posed by Cristopher Moore. Moore had suggested that billiards in sufficiently high dimensions—particularly three-dimensional billiards—could support universal computation, but that low-dimensional systems might fall below the universality threshold.
The consequence is profound. Since the Turing halting problem is undecidable, undecidable trajectories arise in physically natural billiard models: there are questions about the ball’s motion that no algorithm can answer. This is significant because billiards emerge as limits of smooth Hamiltonian systems with very steep confining potentials, as well as exact reformulations of the dynamics of hard-sphere gases. The article also explores a connection with the limits of collision chains in celestial mechanics. The paper was written during Eva Miranda’s stay in Zurich as a Nachdiplom lecturer at ETH’s FIM during the fall semester of 2025.
This work follows the article “Turing Complete Navier–Stokes Steady States via Cosymplectic Geometry,” published in May 2026 in PNAS Nexus by Eva Miranda’s doctoral student Søren Dyhr, together with Ángel González-Prieto, Eva Miranda, and Daniel Peralta-Salas. That article extends universal computing capability to the steady states of the Navier–Stokes equations—in other words, to viscous fluids—using cosymplectic geometry. Both works continue the line of research initiated by “Constructing Turing Complete Euler Flows in Dimension 3,” published in PNAS in 2021. That research demonstrated that ideal fluids in three dimensions can be Turing complete.
E. Miranda and I. Ramos, “Two-dimensional billiards are Turing complete”, Proc. Natl. Acad. Sci. U.S.A. 123 (37), e2614500123 (2026).
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