2020/09/11 by J. Barry Love, Love, Jack
Computer Science · Engineering · Mathematics · #52C25 #Advanced Materials and Mechanics #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #General Topology (math.GN) #Mathematics and Applications #primary: 58D29 #secondary: 58A35
paper · pdf · doi:10.48550/arxiv.2009.05662
openalex publication_date 2020/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A square, in the usual sense of the word, is 2-dimensional no matter the dimension of the ambient space in which it is embedded. If the ambient space has dimension greater than or equal to 3 we may bend the square along a diagonal to produce a 3-dimensional polygon with the same edge-lengths. And yet even if the dimension of the ambient space is large, no amount of bending of the square will produce a polygon of dimension larger than 3. We generalize this idea to show that there are only finitely many moduli spaces of polygons with given edge-lengths, even as the ambient dimension increases without bound.