4-D embedding. No self-intersection in ℝ⁴: blue is positive w, red negative. Drag z–w — the pinch travels; the crossing is only in the shadow.
Kac asked whether you can hear the shape of a drum. Honest answer: it’s hard — the two drums share their deepest partials, so they sound like cousins. The difference is a family of modes that exist only because of the flip: waves that close up by crossing the twisted seam mirrored, ringing between the shared partials at √5⁄2, √13⁄2… of the fundamental. Blue — in the ripples and the numbers below — is what both drums share; red is the twist’s own voice, and striking the torus never makes red. Compare drums plays the same strike on both, back to back; isolate the twist plays only what the flip adds.
This little square is the whole site in miniature: blue edges glue straight, red edges glue with a flip. The letter crosses a red edge and returns as its mirror image — and its path slowly weaves the square, blue strands walked right-handed, red mirrored. With sound on, its rising motif comes back falling at each flip. Click to steer the weave.
Steering is the snake’s own left and right — and the snake is chiral. Cross the red seam and it returns mirrored, so your left turns bend right on screen until you cross again. Eating strikes the drum at that point. Arrow keys or A/D while playing; Escape pauses.
Lines wrap through the gluings; on the unrolled plane every second row of boards is a mirror image, and winning lines run straight through it. The flip gives this board 19 ways to win to the torus’s 12. A hat-tip to Jeff Weeks’ Torus Games.