We study the ferromagnetic Ising model on a two-dimensional transition lattice that continuously interpolates between the Lieb and kagome lattices via a geometric deformation parameterized by the angle $\theta$. The exchange couplings follow a distance-dependent decay controlled by an exponent $n$, enabling us to disentangle the interplay between geometry and interaction range. Using extensive Monte Carlo simulations and Binder cumulant analysis, we map the phase diagram $T_c\left(\theta \right)$ and show that the critical temperature increases monotonically as the lattice acquires higher effective connectivity. In the large-$n$ limit, the system approaches a nearest-neighbor regime in which the Monte Carlo critical temperatures converge to the known exact values of the Lieb and kagome limits. Finite-size scaling and data collapse confirm that the entire interpolation belongs to the two-dimensional Ising universality class. Beyond demonstrating the robustness of the critical behavior across the Lieb-kagome transition, the present framework suggests natural extensions. The continuous geometric interpolation between the nonfrustrated Lieb lattice and the frustrated kagome lattice offers a suitable platform to explore antiferromagnetic interactions, where the interplay between frustration and long-range couplings may induce nontrivial ordering.

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• Continuous phase transition
• Critical exponents
• Phase diagrams
• 2-dimensional systems
• Kagome lattice
• Ising model
• Lattice models in condensed matter
• Monte Carlo methods