A piecewise-linear isometrically immersed flat Klein bottle in Euclidean 3-space

2025/04/10 by Stepan Paul, Paul, Stepan · 2 voices
Engineering · Physics and Astronomy · #Advanced Materials and Mechanics #Structural Analysis and Optimization #Micro and Nano Robotics

paper · pdf · doi:10.48550/arxiv.2504.08826

Abstract

We present numerical polyhedron data for the image of a piecewise-linear map from a zero-curvature Klein bottle into Euclidean 3-space such that every point in the domain has a neighborhood which is isometrically embedded. To the author's knowledge, this is the first explicit piecewise-smooth isometric immersion of a flat Klein bottle. Intuitively, the surface can be locally made from origami and but for the self-intersections has the global topology of a Klein bottle.

Discussions

• The second talk is by Stepan Paul, who tells us about "Topological Origami: Beyond the Paper Donut."🍩Flat tori are known, but what about a flat Klein Bottle? Stepan shows us a first prototype (that f [bsky, 5 points, 0 comments]
• "A piecewise-linear isometrically immersed flat Klein bottle in Euclidean 3-space", or, in other words, an Origami Klein bottle with self-intersections: https://arxiv.org/abs/2504.08826 [bsky, 1 points, 0 comments]

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