Abstract: Many iterative algorithms can be thought of as fixed-point iterations of contractive or nonexpansive operators. Traditionally, such algorithms and operators are analyzed analytically, with inequalities. Since Eckstein and Bertsekas (Figure 1, Math Program 55:293-318, 1992), circles and half-spaces have been used to geometrically illustrate operator theoretic notions although the actual analyses, proofs, and the computation of optimal stepsizes were done analytically with inequalities. In this talk, we formalize a correspondence between common operators (such as proximal mapping and subdifferentials of convex functions) and geometric objects on the complex plane. We use elementary Euclidean geometry to rapidly prove many useful results regarding the convergence of fixed-point iterations and their optimal stepsizes. The formalism maps various classes of operators to sets on the complex plane and also maps algebraic operations such as scaling, inversion, addition, and composition of operators to geometric operations on sets on the complex plane. Equipped with these tools, we use geometric arguments to review classic results and obtain two novel convergence results. This talk includes joint work with Ernest Ryu and Robert Hannah (arXiv:1902.09788) and with Xinmeng Huang and Ernest Ryu (arXiv:arXiv:1912.01593).