Möbius Strip Unrolled (2D)

Half-Twists

Ant Motion

Crawl speed 1.0x

Pick a half-twist count, then press Crawl.

Live Readout

Loops completed

Side / face

Face A

Orientation

Orientable

Position u

0°

How it works

This diagram "unrolls" the strip along its length: the horizontal axis is the loop position u (0° to 360°, then wrapping), and the ant's vertical wobble traces cos(n·u/2), where n is the number of half-twists. That factor is exactly the half-angle term in the real parametric Möbius surface, x=(R+v·cos(u/2))cos(u).

n = 0 — a plain closed band (a cylinder unrolled). No twist, so the ant's face never flips: after one loop it is back on the same face it started on. Orientable.
n = 1 — the classic Möbius strip. The face flips once per loop, so after one lap the ant is on the "other side" — which is really the same single side, since the strip has only one. It takes two full loops to return to the exact start. Non-orientable.
n = 2 — two half-twists. The face flips twice per loop, cancelling out, so the ant is back on its starting face after just one loop even though the strip looks twisted. Orientable, like n = 0.

The fading trail left behind shows the same thing geometrically: with an odd twist count the second loop's trail is a mirror image of the first, interleaving to cover the whole band — proof the surface has a single continuous side and a single boundary edge.

Face A Face B Ant