Flat Klein Bottles

Geometry and Topology Preserving Realizations of the Flat Klein Bottle

Below are two models I have made of “flat” Klein bottles—ones that can be folded from paper but for their self-intersections—that I discovered.

The three-cylinder model

A curved-crease origami model of piecewise-smooth path isometric image of a flat Klein bottle.

• Bridges paper
• Interview with Physics Magazine
• Crease pattern

The polyhedral model

An origami model of a flat immersed polyhedral Klein bottle that appeared in the Maison Poincaré in Paris. Photo credit: Edmund Harriss.

• arXiv paper
• Interactive 3D viewer
• Crease patterns. Three of each pattern are needed: straights, joints.

More

• An interactive paper torus.