The Flat Klein Bottle Rendered in Curved-Crease Origami

Abstract

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.