Multigeometries

Multigeometries are mathematical structures that endow a common set with multiple geometric frameworks. Given a set X and a family {G_i} of geometric structures indexed by I, a multigeometry studies how these different geometries coexist on X and interact. The goal is to understand how distance, incidence, lines, and other geometric notions from the various G_i influence one another and what new properties emerge from their combination.

A formal definition typically treats a multigeometry as a tuple (X, {G_i}_{i in I}, C), where each G_i specifies relations among points, lines, or higher-dimensional flats, and C encodes compatibility rules between the structures. Depending on the context, the G_i may be incidence geometries, metric spaces, projective or affine geometries, or combinatorial abstractions such as matroids. The compatibility conditions ensure that the different geometries share a coherent interpretation on X.

Examples arise by pairing familiar geometries on the same point set, such as affine and projective geometries with a prescribed alignment of incidences, or by combining multiple metrics on X. Multigeometries have potential applications in computer graphics, where multiple space models are used, and in data analysis, where different relational notions capture distinct structure. The notion is not universally standardized, and definitions vary across fields, but the concept provides a common framework for studying systems that admit more than one geometric viewpoint.