If M is orientable (that is, two-sided), the parallel surface has
two components. On the other hand, if M is nonorientable (that is, one-sided) the parallel surface Mt is connected. Furthermore, when M is nonorientable the parallel surface Mt is orientable. The purpose of this notebook is to illustrate these facts is the case of one of the simplest nonorientable surfaces, the Möbius strip.
To describe the parallel surface Mt, we need a Mathematica
operator parsurf that produces from a parametrization x of M a parametrization parsurf[x][t] of Mt . Since we may want to draw surfaces at different distances t from M, we should include t as a variable. The following package Parallel.m does the job:
This command may be broken down as follows. First one defines the tangent vectors xu and xv. Then n1 is the cross product of xu and xv, and n4 is a unit length vector pointing in the same direction as n1. For fixed t the surface Mt has the parametric representation
Usually the command ParametricPlot3D, when used to plot surfaces, takes as its argument a 3-tuple. However, it can also take a 4-tuple as an argument. In this case, the fourth entry in the argument specifies how to color the surface. The following command creates from Mathematica's parametric representation of a surface a 4-tuple containing a color specification:
A more informative picture can be obtained using the graphics command FaceForm to color different sides of the polygons that Mathematica uses to approximate a surface. To use FaceForm we replace color by FaceForm[color1,color2]. Thus, for example,
Since the Möbius strip is nonorientable, any attempt to cover all the backs and fronts of polygons consistently must fail; in other words, the two colors must meet.
On the other hand, since a parallel surface to a Möbius strip is orientable, it is possible to color its back and front with different colors in order to distinguish the two sides of the surface parallel
to the Möbius strip. The four-color version of a Möbius strip and one of its parallel surfaces is generated by the command