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

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.

Figures (1)

• Figure 1: The image of a piecewise linear local isometric-embedding of a flat Klein bottle into $\mathbb{R}^3$.