Vision Transformers have achieved remarkable success in computer vision, but their common use of learnable one-dimensional positional encodings weakens the inherent two-dimensional spatial structure of images after patch flattening. Existing positional encodings often lack geometric constraints and do not preserve a monotonic relationship between Euclidean spatial distances and sequential index distances, limiting ViTs' ability to exploit spatial proximity priors. Motivated by the usefulness of periodicity in positional encoding, we propose Weierstrass elliptic Positional Encoding (WePE), a mathematically grounded method for encoding two-dimensional coordinates in the complex domain. WePE maps normalized 2D patch coordinates onto the complex plane and constructs compact four-dimensional positional features using the Weierstrass elliptic function and its derivative. The double periodicity provides a principled representation of 2D positions, and its intrinsic lattice structure naturally matches the regular geometry of image patch grids. Its nonlinear geometric properties help model spatial distance relationships more faithfully, while the algebraic addition formula enables relative positional information between arbitrary patch pairs to be derived directly from their absolute encodings. WePE is plug-and-play and resolution-agnostic, allowing seamless integration into existing ViTs. Extensive experiments show that WePE brings consistent performance gains in most settings. With precomputed lookup tables, these improvements introduce no noticeable computational or memory overhead. Additional analyses and ablation studies further validate the effectiveness of the proposed method.