The previous page presents a flat Möbius band, but it’s also possible to embed the Möbius band cleanly (without self-intersection) in three-dimensional space.
A surface lying in a three-dimensional space is said to be one-sided if it’s possible for an ant to travel on it and arrive on the opposite side from where it started. In contrast to cylinders, spheres, and tori which are two-sided, Möbius bands are one-sided. We can use our program to see this in two ways.
First, select Mobius band and choose two different colors under color1 and color2 (say blue and white). Notice that the blue and white strips meet at two edges spanning the band’s width, and that each of these strips covers half the Möbius band’s total surface. 1 In contrast, the cylinder is also painted with both blue and white strips but they don’t meet; each covers only one side of this two-sided surface.
We can also see the Möbius band’s one-sidedness by following the animated ball’s path, which traverses the entire surface. In contrast, in the cylinder, the ball traverses only one side of the surface without crossing to or reaching the other side.2
Interestingly, the Möbius band has only a single edge that forms a loop. Its length is twice that of the band. In the program, place the ball along the edge by setting v to 0.1 or 0.9.3 Halfway along its journey, the ball arrives at a point opposite its starting point, and it completes its journey back at its starting point.
- In the flat Möbius band represented by a rectangle with a pair of glued edges, this corresponds to painting one side of the rectangle blue and the other side white. ↩
- See if you can get the ball to roll along the cylinder’s ‘inner’ side and also its ‘outer’ side. ↩
- The slider v sets the ball’s position along the band’s width. ↩