2025/09/23 by Zou, Zhengyu
Engineering · Mathematics · #51M10 #52B70 #57Q15 (Secondary) #57Q35 (Primary) #Advanced Materials and Mechanics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2509.18668
openalex publication_date 2025/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that there exists a geodesic triangulation T of a hyperbolic genus 2 surface Σ2 with 10 vertices and an isometric polyhedral embedding S: Σ2 \hookrightarrow ℍ3 that sends the triangles in T to geodesic triangles in ℍ3. We call this type of embedding a hyperbolic origami 2-torus. Since 10 is the combinatorially minimum number of vertices required to triangulate a genus 2 surface, this paper settles the question of minimum number of vertices required to obtain a hyperbolic origami 2-torus.