One side, one edge — and cutting it does not do what you expect.
Where's the maths? This is topology — the branch of mathematics that studies what stays true about a shape when you bend and stretch it, but never cut or glue it. A topologist does not care how long the strip is or how tightly it curves; all that matters is how it is joined up. Glue the two ends of a paper strip straight across and you get a cylinder: two sides you could paint different colours, and two separate edge circles. Give one end a half-twist before you glue and you get a Möbius strip, which has only one side and only one edge — the gold dot on screen walks the entire surface and returns on what looked like the opposite face, without ever crossing an edge. Because there is only one side, the ordinary rules break: cutting it down the middle does not give two loops, it gives one loop twice as long. The only thing that decides the answer is whether the number of half-twists is odd or even, so you can predict every result before you cut. The Klein bottle is the same trick one dimension up: a closed surface with no inside and no outside.
Beyond the syllabus: topology (one-sided surfaces, Klein bottle).
The strip is drawn from an exact recipe, not a picture. Walk an angle u once round a circle of radius R, and at each step lay down a short line segment that has itself turned through the angle nu/2. A point of the strip then sits at horizontal distance R + v cos(nu/2) from the centre, in the direction (cos u, sin u), and at height v sin(nu/2) — with v running from one edge to the other. Here n is the half-twists slider. When u reaches 2π the segment has turned through n·180°, so for odd n it comes back upside-down and the two ends glue with a flip.
Odd number of half-twists ⇒ one side and one edge. Even ⇒ two sides and two edges. That single fact explains everything on this screen. For odd n, the boundary is one closed curve of double the length, and the travelling dot needs two laps to get home.
Now cut. The cut line is itself a curve on the surface, and it obeys the same rule — for an odd strip, a cut at ⅓ of the width is one curve that goes round twice. What is left over is what the app builds for real, as separate parametric bands:
| strip | cut down the middle | cut at ⅓ |
|---|---|---|
| 0 twists (cylinder) | 2 loops, free | 3 loops, free |
| 1 twist (Möbius) | 1 loop, 2× long, 4 half-twists | 2 loops, linked |
| 2 twists | 2 loops, linked | 3 loops, linked |
| 3 twists | 1 knotted loop (trefoil) | 2 loops, linked |
Why one loop from the Möbius? The centre line of a Möbius strip is a single closed curve, and the surface only has one side of it — so cutting along it does not divide the strip into two halves. It just unrolls the one piece you had into a longer band with 2n + 2 half-twists: 4 for n = 1.
Symmetry is the one piece of syllabus geometry hiding here. The Möbius strip has a rotational symmetry of order 2 — turn it a half-turn about the axis through its centre and it lands on itself — but no mirror symmetry: a left-handed strip and a right-handed one cannot be laid on top of each other.
Glue two Möbius strips together along their single edges and you get a Klein bottle: a closed surface with no boundary and no inside. It cannot sit in 3-D without passing through itself, so the model on the tab is an immersion — an honest shadow of a four-dimensional object.
Controls: drag to orbit, scroll or pinch to zoom; sliders set the twists, the width and how far the cut pieces are pulled apart; press c to cut and r to rejoin.