We study the Ising model on two-dimensional surfaces discretized using the Fibonacci method with Delaunay triangulation, considering the ring, cut ring, and torus topologies. The phase diagrams reveal a universal critical temperature of TC ≈ 3.33(3)J in the thermodynamic limit, which is consistent with the results for the Fibonacci sphere [1]. Despite the exclusion of topological defects (vertices with coordination numbers 5/7) in the ring and cut ring Fibonacci configurations, deviations from the critical temperature of the ideal flat triangular lattice are observed. The TC values, similar to the spherical case, experience shifts. Notably, the torus, which possesses the minimal defect density (