Riemann surface of the torus

A Riemann surface of a torus is a way mathematicians visualize complex functions on a shape similar to a doughnut. It transforms the torus into a smooth, multi-layered surface where complex analysis becomes clearer, especially for functions that are complicated on the original shape. Imagine wrapping and flattening the torus in a way that allows complex functions to be studied without interruptions caused by holes or overlaps. This approach helps understand how these functions behave globally, providing insights into advanced areas like algebraic geometry and complex analysis through a geometric perspective.