Moebius Strip

Additional images: Anaglyph, Anaglyph Wireframe, Parallel Stereo
LiveGraphics3D, JavaView
                   The Mobius Strip

                   Hermann Karcher

  Parametric Equations

  x = aa * (cos(v) + u * cos(v / 2) * cos(v))
  y = aa * (sin(v) + u * cos(v / 2) * sin(v))
  z = aa * u * sin(v / 2)

The Mobius Strip is perhaps the most famous of the one-sided or 
"non-orientable" surfaces.

A Mobius Strip can be found on any non-orientable surfaces. To
see one on the Klein Bottle, select from the Settings menu
"Set t,u,v Ranges" and put  umin = - 0.4,  umax = + 0.4  .
On the Boys surface, there are even two different kinds. To see
one with three half-twists, change to  umin = 0.9; this is a
band with the "equator" of the Boys surface as its centerline.
Bands with meridians as center curves are ordinary Moebius bands.
To see one, change the u,v-ranges to umin = -0.998,  vmin = 6.1 : 

(On the Steiner Surface and on the Crosscap, one can also find
Mobius Strips. However these are not embedded and so are not 
easily recognizable. In math jargon, these surfaces are not 
immersed and there is a singularity on the Moebius band.)

MoebiusToKlein.mov (700 k) This movie shows closing the edge of a Moebius Strip, which results a Klein Bottle.


Supporting files: Description in PDF