Penrose Tiling on Curved Spacetime: A Unified Framework for Geometry, Topology, Quantum Information, and Real-World Applications

This article presents a comprehensive framework integrating Penrose tiling with the curvature of spacetime, leveraging differential geometry, computational algorithms, and interdisciplinary applications. We use the Gauss–Bonnet theorem to govern tile distribution on non-Euclidean surfaces, extending this to 4D Lorentzian manifolds for spacetime applications. Novel contributions include a detailed tiling algorithm with pseudocode, higher-dimensional embeddings for simplified computation, and analogies to quantum error-correcting codes. Applications span puzzle-solving, protein folding, virus capsid modeling, quantum gravity, and quantum information safeguarding. This framework provides both theoretical insights and practical tools.