Geometry

Abstract: The Plateau problem is a fundamental challenge in geometric measure theory with the objective of finding minimal surfaces that have prescribed boundaries. Originating from Joseph Plateau's 19th-century soap film experiments, this problem seeks to identify surfaces that locally minimize surface area, with natural occurrences observed in membranes under equal opposing pressure, exemplified by soap films spanning wireframes. We focus on minimal surfaces within three-dimensional space, specifically exploring a unique subset called minimal mod-3 surfaces. These intriguing surfaces comprise oriented minimal surface pieces meeting in groups of three along "singular curves," with consistent 120-degree angles between intersecting faces. Importantly, they possess a unique characteristic: their orientations allow their boundaries to cancel out modulo 3. Our research adopts an innovative wireframe approach to address this problem, focusing on wireframes designed with hexagonal symmetry, allowing the extension of minimal-area surfaces across three-dimensional space through rotations and translations. The result is the creation of a "hexaprism" surface featuring both hexagonal symmetry and minimal mod-3 characteristics.