The surface
4-D embedding. Here the surface lives in ℝ⁴ with no self-intersection at all: teal marks points with positive w-coordinate, ember negative. What you see is its 3-D shadow. Drag the z–w slider and watch the pinch line travel around the ring — the apparent self-crossing is an artifact of projection, like the crossings in a drawing of a knot.
The drum
In 1966 Mark Kac asked whether you can hear the shape of a drum. Here you can try: this square is a real drumhead whose partials are the true vibration frequencies of the surface it's glued into. Strike it and it rings. The torus (both edge pairs glued straight) allows every sine and cosine mode. The Klein bottle's twisted gluing censors half of them — a mode survives only if its shape agrees with its own mirror image across the flip — and admits others the torus cannot hold, at frequencies like √5⁄2 of the fundamental. Same square, same area, different chord. The pattern you see ringing is the actual sum of the excited modes.
Life on the surface
Cut a Klein bottle open and it flattens to this square: the teal edges glue side-to-side normally, but the ember edges glue top-to-bottom with a flip. That one flip is the whole difference from a torus. Watch the letter cross an ember edge: it comes back as its mirror image, and nothing but a second crossing will un-mirror it. With sound on, the walker hums a rising motif — which comes back falling after the flip. Chirality you can hear.
Games on the surface
Snake, non-orientable
Steering is the snake's own left and right — and the snake is a chiral thing. Cross the twisted edge and it returns mirrored, so your left turns now bend right on screen. That isn't an added gimmick; it is exactly what the surface does to anything with a handedness. Eating strikes the drum at that point. Arrow keys, A/D, or the buttons.
Tic-tac-toe through the twist
Rows and columns wrap through the gluings, so lines that look broken on the board run perfectly straight on the unrolled plane at right — where every second row of boards is a mirror image, because that is how the Klein bottle tiles the plane. A column that leaves the top re-enters the bottom at the mirrored column, so the twist adds lines a torus doesn't have: this board has 19 ways to win to the torus's 12. Wraparound boards favor whoever moves first — try giving the computer the first move.