### Building Moebius Bands

We are all familiar with the Möbius strip or Möbius band. This topologically intriguing object with one side and one edge has fascinated children of all ages since it was discovered independently by August Möbius and Johann Listing in the same year, 1858.

Building the Band

It is a simple matter to make a Möbius band from a long rectangular strip of paper. Here we are concerned with the geometrical construction of the surface. We start with a circle, and a small line segment with centre on this circle. The segment may be in the plane of the circle or perpendicular to it.

Now as we move the segment around the circle, it generates a surface. If the segment remains in the plane of the circle, we get an annulus. If it remains perpendicular to it, we get a cylinder:

A half-twist of the Segment

If we rotate the line segment about the tangent as it is moved around the circle, a variety of surfaces may be generated. With a half-twist (180°) in one rotation around the circle, the surface corresponding to a Möbius band is generated.

Let us colour one end of the line segment red and the other magenta. Each end traces a curve during the process of generation. But the two curves join to form a single continuous curve, homeomorphic to a circle:

More Twists

If the segment turns through a full twist (360°) as it moves around the circle, the boundary curve comprises two inter-linked closed curves. This pair is called a Hopf link.

With three half-turns (540°) another one-sided surface with one edge is formed. The edge is now knotted: while it is homeomorphic to a circle, it cannot be continuously deformed into a circle without self-intersection. It is not ambient-isotopic to a circle; it is a trefoil knot.

Many more interesting surfaces can be generated. Some have two sides and others have only one. The two-sided surfaces have two linked boundary curves while the one-sided surfaces have a single edge which is knotted. A few of these are shown here:

The Equations

The process of moving the line segment around the circle leads us to the equations for the Möbius band. In cylindrical polar coordinates the circle is ${(r,\theta,z) = (a,\theta,0)}$. The tip of the segment, relative to its centre, is ${(r,\theta,z) = (b\cos\phi, 0, b\sin\phi)}$ where ${b = \frac{1}{2}\ell}$ is half the segment length and ${\phi = \alpha\theta}$, with ${\alpha}$ determining the amount of twist. Thus, the tip of the line has ${r=a+b\cos\alpha\theta}$ and ${z=b\sin\alpha\theta}$.
In cartesian coordinates, the equations become

$\displaystyle \begin{array}{rcl} x &=& \left( a + b\cos\alpha\theta \right) \cos\theta \\ y &=& \left( a + b\cos\alpha\theta \right) \sin\theta \\ z &=& \left( b\sin\alpha\theta \right) \end{array}$

These are the parametric equations for the twisted bands, with ${\theta \in [0,2\pi]}$ and ${b\in [-\ell,\ell]}$. For the Möbius band, ${\alpha=\frac{1}{2}}$.