Hopf Fibrations, 2022

The Hopf Fibrations study investigates three-dimensional visualizations of higher-dimensional topology. The Hopf fibration — a mapping from the 3-sphere to the 2-sphere — translates into geometric patterns through physical models. Circles in four-dimensional space project into three dimensions as interlocking curves, where each fiber creates distinct geometric paths that link with every other fiber without intersection. Stereographic projection methods transform higher-dimensional geometry into printable three-dimensional forms. The choice of projection point and fiber subset determines which topological relationships remain visible in the physical object. Fiber density and tube radius constrain the printable output: too many fibers produce unprintable intersections, too few lose the structural pattern. Different visualization methods are tested for how effectively they communicate the fibration's structure and how projection parameters affect pattern clarity. Physical models demonstrate fiber linking patterns and the non-trivial topology underlying these mathematical structures. The investigation remains ongoing.