### Moebiquity: Ubiquity and Versitility of the Möbius Band

The Möbius strip or Möbius band, with one side and one edge, has been a source of fascination since its discovery in 1858, independently by August Möbius and Johann Listing. It is easily formed from a strip of paper by giving it a half-twist before joining the ends.

Möbius band in 3-space and a flat representation in 2-space.

The Möbius band may be characterised in many ways:

• The set of chords in the unit circle.
• The configuration space of two unordered points on a circle.
• A projective plane with a disk removed.
• The set of tropical diameters of the Earth.
• The set of lines through the centre of a (finite) cylindrical tube.
• The topological quotient of a torus by a group action.
• The simplest nonorientable surface.

Applications

The Möbius band occurs widely in mathematical art. It is used in the design of necklaces, brooches, scarfs, etc. In music theory, the space of all two-note chords (dyads) has the form of a Möbius band. For more general chords with more than two notes, higher-dimensional counterparts of the Möbius band, known as orbifolds, are used.

There have been many technical applications of Möbius bands. They have been used as conveyor belts: the entire surface area of the belt gets the same amount of wear, giving the belts longer lifetimes. Möbius bands have also been used in continuous-loop recording tapes, doubling the total playing time. They are also used for computer printer and typewriter ribbons, as the ribbon is twice as wide as the print head and both halves are worn evenly. There are also many applications in physics, chemistry and electronics.

The Flat Earth

The surface of the Earth is well approximated by a sphere. The topology of this closed 2-dimensional surface can be represented by a square in which we identify pairs of points on the boundary. To start with a simpler example, take a (finite) cylinder, cut along a line parallel to the axis and unroll to a flat rectangle. Then mark the cut edges to indicate that they represent the same set of points. We get a flat representation of the cylinder. A point moving out through the right edge appears at the left edge (see Figure, top left).

Flat representations (in 2D) of surfaces in 3D (or 4D) space.

The sphere corresponds to the top-centre panel in the Figure. This representation might serve well as a logo for the Flat Earth Society. There is a map projection, the Collignon projection, that represents the surface of the earth within a rhombus. The left and right edges both correspond to the international date line (180 degrees East and West) and represent the same set of points. This is structurally similar to the diagram in the previous figure.

Map of the Earth using a Collignon projection.

Other flat representations of 2-dimensional curved surfaces are shown in the Figure above. The bottom right panel shows the Möbius band, in which the two sides ${\mathbf{AB}}$ are to be joined together preserving the orientations shown.

Folding a Torus to get a Möbius Band

The torus is the configuration space of all ordered pairs of points on the circle ${\mathbb{S}^1}$. A torus can be constructed as the square ${[0, 1]\times[0, 1]}$ with the edges identified by ${(0, y) \sim (1, y)}$ (identify left and right sides) and ${(x, 0) \sim (x, 1)}$ (identify the bottom and top edges). We denote the torus as ${\mathbb{T}^2 = \mathbb{S}^1\times\mathbb{S}^1}$.

Topologically, the Möbius strip can be defined as the unit square ${[0, 1] \times [0, 1]}$ with its left and right sides identified by ${(0, y) \sim (1, 1-y)}$ for ${0 \le y \le 1}$. [It is almost, but not quite, a product of a circle and an interval, ${\mathbb{S}^1\times\mathbb{I}}$ (this is actually a cylinder)].

Constructing a Moebius band by folding a torus.

To convert the torus into a Möbius band, we fold along a diagonal and identify coincident points. The resulting triangle is disected along the line shown (to be rejoined later), The triangle is split into Part~1 and Part~2. Part~1 is turned over and moved to the right as shown and the two parts are joined along the blue arrows. The result (bottom right panel) is the diagram of a Möbius band.

The Möbius strip is also the topological quotient of a torus by a group. If for the torus we assume ${(x, y) \sim (y, x)}$, we obtain a Möbius strip. The diagonal of the square becomes the boundary of the Möbius strip. It carries an orbifold structure and is denoted ${\mathbb{T}^2/\mathbf{Z}_2}$, the quotient of a 2-torus ${\mathbb{T}^2}$ by the action of ${\mathbf{Z}_2}$, the symmetric group of order 2. It is also the configuration space of pairs of points on the circle (disregarding order). The Möbius strip is the simplest example of a surface that is not orientable.

Connection with the Real Projective Plane

The real projective plane ${\mathbf{RP}^2}$ can be represented by all (undirected) lines through the origin of ${\mathbb{R}^3}$. Equivalently, it corresponds to the set of all the diameters of ${\mathbb{S}^2}$. Another way of visualising ${\mathbf{RP}^2}$ is as the set of all (unordered) pairs of antipodal points on the 2-sphere.

Now imagine removing a northern polar cap from the globe. Since diametrically opposite points are identified, the corresponding southern cap also goes. What remains is a tropical zone. The set of diameters in this zone has the topology of the Möbius band.

Removing a disk from the projective plane to obtain a Moebius band (image from Weeks, 2002).

Thus, a Möbius band is equivalent to the real projective plane with a disk removed.

Sources

Weeks, Jeffrey R., 2002: The Shape of Space. Second Edn., CRC Press, 382pp.