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We give a brief review of 2D, Euclidean, centroidal Voronoi tessellations (CVTs). Dropping to the specific case of two-point CVTs, a rich interaction between calculus, geometry and analysis is uncovered. Narrowing further to “ellipse-like” shapes with order 2 rotational symmetry, a full characterization of possible two-point CVTs is given for these figures. One might expect the CVTs of symmetric figures with even rotational symmetry to always correspond to lines of symmetry of the figure. Surprisingly, this conjecture is false. We close the article showing how to generate counterexamples to this conjecture and provide some direction for further inquiry.

Two-Point Centroidal Voronoi Tessellations