# How does one plot a Möbius Strip?

How can I plot a Möbius strip? I tried this one, but I can't get it to work:

ParametricPlot3D[{
(5 + s*Cos[u/2]) Cos[u],
(5 + s*Cos[u/2]) Sin[u],
(s*Sin[u/2])}, {u, -20, 20}, {s, 0, 2 π}]


• Please post the code that you used that didn't work for you. Without the code, it's impossible for us to tell you what you did wrong. Sep 18, 2015 at 10:37
• ParametricPlot3D [[[5 + sCos[u/2]]*Cos[u], [5 + sCos[u/2]]*Sin[u], s*Sin[u/2]], {u, -20, 20}, {s, 0 , 2 [Pi]}] @Pickett Sep 18, 2015 at 10:41
• You need at least a space or a * between s and Cos and Sin. There are two instances where you forgot that. Sep 18, 2015 at 10:49
• Just for fun: Here's an animation I made last year. Sep 18, 2015 at 18:05
• See also the attempted solution in the question Cutting bagels into linked halves.
– Jens
Sep 18, 2015 at 20:16

## 3 Answers

Equation taken form the wiki page

 x[u_, v_] := (1 + (v/2) Cos[u/2]) Cos[u]
y[u_, v_] := (1 + (v/2) Cos[u/2]) Sin[u]
z[u_, v_] := (v/2) Sin[u/2]

plot = ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, 0,
2 Pi}, {v, -1, 1}, Boxed -> False, Axes -> False]


Just for fun (using the parametrization from OP hyperlink):

mobius[r_, s_,t_] := {r + s Cos[t/2], r + s Cos[t/2], s Sin[t/2]} {Cos[t], Sin[t],
1}
Manipulate[
With[{wd = w},
Row[{
ParametricPlot[{u, v}, {u, -wd, wd}, {v, 0, 2 Pi},
AspectRatio -> 1/2, Epilog -> {Red, PointSize[0.04], Point@p},
ImageSize -> 200],
Show[ParametricPlot3D[mobius[r, u, v], {u, -wd, wd}, {v, 0, 2 Pi},
Mesh -> False, PlotStyle -> Yellow,
PerformanceGoal -> "Quality"],
Graphics3D[{Red, PointSize[0.04], Point[mobius[r, ##] & @@ p]}],
ImageSize -> 200]
}]], {r, 1, 2}, {w, 0.5, 2}, {{p, {0, 0}}, {{-w, 0}, {w, 2 Pi}},
Slider2D}]


• computerbasedmath.org, animating math to make it more intuitive. Sep 18, 2015 at 17:42

...and, just for the sake of variation, here is a "minimal" Möbius strip (that is, a minimal surface with the topology of a Möbius strip). This example is due to Meeks.

ParametricPlot3D[{(r^3 - 1/r^3) Cos[3 θ]/3 + (r^2 + 1/r^2) Cos[2 θ] + (r - 1/r) Cos[θ],
(r^3 - 1/r^3) Sin[3 θ]/3 + (r^2 + 1/r^2) Sin[2 θ] + (r - 1/r) Sin[θ],
2 (r - 1/r) Sin[θ]}, {r, 1, 4/3}, {θ, -π, π}]