Two Full Twists Are Nothing

A clamped ribbon with one full twist can never be flattened — with two it melts to dead flat before your eyes. In rotation-land, 720° is the new 0°, and the invariant is a single bit.

mode

A striped belt between two steel clamps. The rules: while SETTING UP, the right clamp may turn to load twists — but once play begins, a clamp may only TRAVEL: up, over, around, never rotating. The teal arrow rides the right clamp; if it turns mid-game, someone cheated.

Step 1 of 6 — House rules

Hold a plate flat on your palm and rotate it 360° — your arm ends up twisted. Rotate it 360° more, the same way, and your arm comes back to normal: the famous plate trick. The belt is the honest version: the twist you can never remove with one turn vanishes completely after two.

The demo's whole claim fits in one line: with ends clamped, twist is only conserved mod 2 — going around the body once trades exactly two twists for a trip, and nothing trades one.

References: P. A. M. Dirac's string trick (c. 1929), via M. Gardner, Riddles of the Sphinx · L. H. Kauffman, Knots and Physics, on belts, framings and the plate trick · π₁(SO(3)) ≅ ℤ/2, any algebraic topology text.