Möbius Strip
Shape · (x, y, z) = f(u, v, t)
((2 +(2v/π − 1)/2 · cos(u/2)) · cos(u), (2v/π − 1)/2 · sin(u/2), (2 +(2v/π − 1)/2 · cos(u/2)) · sin(u))
Open in the app The dials and keys named below are the app's.
What it draws
The parameter u goes once round a circle of radius 2 lying in the ground plane, and v runs across the band. The expression (2v/π − 1)/2 does that crossing. At Span of v (×π) 1 the parameter v runs 0 → π, so the bracket runs −1 → 1 and the half of it runs −0.5 → 0.5. That is the signed distance from the centre line, and the band is one unit wide.
The second expression is the height, (2v/π − 1)/2·sin(u/2), so the band never rises more than 0.5 above the floor. Nothing depends on t, so it stands still.
The half twist
Everything hangs on the halved angle. As u makes one full lap, u/2 makes only half of one, so the band turns through 180° about its own centre line on the way round. At u = 0 it lies flat, its two edges at radius 1.5 and 2.5. At u = π it stands on edge, both edges at radius 2 and at heights ±0.5.
Back at u = 2π it is flat again, but the edge that started at 1.5 is now at 2.5: the two have swapped places. Following the surface all the way round therefore lands on the other face of the strip, and the two faces are the same face. A Möbius strip is non-orientable, it has one side, and its boundary is a single closed curve rather than two rims.
History
The strip is credited independently to two German mathematicians in 1858, Johann Benedict Listing and August Ferdinand Möbius. Both came to it while working out how surfaces can be cut and glued.
Try
- Set Turns of u to 2. The band is drawn a second time over the first, with its two edges exchanged: going round once is not enough to come back to where you started.
- Change every u/2 to 3u/2, as in (2 + (2v/π − 1)/2·cos(3u/2))·cos(u): three half twists per lap, still one-sided.
- Change every u/2 to u instead: a whole turn per lap, and the surface closes into an ordinary two-sided band with two rims.
- Top in the deck: from above the band is widest at u = 0, from radius 1.5 to 2.5, and narrows to the single circle of radius 2 where it stands on edge.