Möbius Strip & Klein Bottle

Drag to rotate; use Ctrl or ⌘ with the wheel to zoom. The mesh controls change only the display resolution.

Enable the orientation path and use Play or the slider. The yellow point follows a closed curve; the arrow transports a local normal. At the starting point it is reversed after one turn and restored after two. It is a geometric demonstration, not a physical ant simulation.

The Möbius strip has one boundary; the Klein bottle has none. Both are non-orientable. Klein-bottle self-intersections in 3D do not connect the crossing sheets. The classic immersion is piecewise and may have visible creases; the figure-eight immersion is smooth. The identified parameter edges are (0,v) ≡ (2π,−v) for Möbius and figure eight, and (0,v) ≡ (2π,π−v) for the classic Klein bottle (angles modulo 2π).

After the first render, valid changes update the view automatically. Mesh resolution and wireframe affect only the drawing. Dragging rotates; Ctrl/⌘ + wheel zooms, and plain wheel scrolls. Reset fits the whole surface and arrow to the available area and returns the path to its start, keeping inputs. PNG saves the current view and pauses playback. Only raw main inputs persist, including inactive or invalid drafts; navigation and reload wait for Render. Camera, path position and playback are not saved. The yellow curve, marker and arrow overlay the surface so they remain visible.