Klein Bottle
Shape · (x, y, z) = f(u, v, t)
((2 + cos(u/2) · sin(v) − sin(u/2) · sin(2v)) · cos(u), sin(u/2) · sin(v) + cos(u/2) · sin(2v), (2 + cos(u/2) · sin(v) − sin(u/2) · sin(2v)) · sin(u))
Open in the app The dials and keys named below are the app's.
What it draws
This is the figure-eight form of the Klein bottle. A tube of centre radius 2 runs round the vertical axis with u, and v runs round its cross-section. The second expression is the height, sin(u/2)·sin(v) + cos(u/2)·sin(2v). Its partner inside the brackets of the other two, cos(u/2)·sin(v) − sin(u/2)·sin(2v), is how far the point lies outside or inside the centre circle. Nothing depends on t.
Take u = 0 to see the cross-section bare. It is (sin(v), sin(2v)), the 1:2 Lissajous figure, a figure eight two units across and two tall, so there the tube spans radius 1 to 3. Once the eight starts turning its reach grows to 1.25 either way, so the surface runs from radius 0.75 to 3.25. The crossing point of the eight sits exactly on the centre circle of radius 2.
Why it has no inside
The halved angle u/2 turns the whole figure eight as the tube travels: half a turn for each full lap. After one lap, at u = 2π, the eight comes back rotated by 180°. For this curve that is the same eight traced with v running backwards, so the surface closes, but only after a reversal.
That reversal is the Klein bottle: a closed surface with no inside and no outside, non-orientable, made by gluing the ends of a band with a flip. A Möbius strip is glued the same way, but here both edges are joined too. No such surface fits in three dimensions without passing through itself, and here it does so along the centre circle of radius 2. The familiar tapering-neck picture is a different way of pushing the same abstract surface into space, and neither picture is the surface itself.
History
Felix Klein described the surface in 1882.
Try
- Change every u/2 to u, as in (2 + cos(u)·sin(v) − sin(u)·sin(2v))·cos(u): a whole turn per lap instead of half, and the surface closes with no flip, so it is two-sided.
- Replace every sin(2v) with cos(v). The cross-section becomes a circle, the reversal stops mattering, and what is left is an ordinary torus of radii 2 and 1.
- Change both 2 + to 3 + to carry the tube round a wider circle: the hole in the middle opens out to radius 1.75.
- Push Mesh to its top stop for a lit solid, where the self-intersection along the centre circle is easiest to find.