2021/01/19 by Stepan Paul, Paul, Stepan
Computer Science · Engineering · #Advanced Materials and Mechanics #Architecture and Computational Design #Differential Geometry (math.DG) #FOS: Mathematics #History and Overview (math.HO) #Interactive and Immersive Displays #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2101.08865
openalex publication_date 2021/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a simple and concrete way of visualizing in three dimensions a "flat" Klein bottle -- one whose local intrinsic geometry is the same as that of a flat plane -- which preserves most its topological and geometric structure. Concretely, the flatness property means that a small patch of the surface around any point can be flattened to a patch of the plane without stretching or compressing. Thus we can use the medium of curved-crease origami with inelastic film to make a model which, except for its self-intersections, necessarily has the flatness property, even along its folded edges. As such, the sculpture presented here illustrates both the flatness, and, through its coloring, the non-orientability of a Klein bottle.