Cross-cap

In mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle. This cannot be done in three dimensions, without the surface intersecting itself.

A cross-cap that has been closed up by gluing a disc to its boundary is called a real projective plane. Two cross-caps glued together at their boundaries form a Klein bottle. An important theorem of topology, the classification theorem for surfaces, states that all two-dimensional nonorientable manifolds are spheres with some number of 'handles' and at most two cross-caps.

Cross-cap model of the real projective plane

A cross-cap can also refer synonymously to the closed surface obtained by gluing a disk to a cross-cap. This surface can be represented parametrically by the following equations:

<math> X(u,v) = r \, (1 + \cos v) \, \cos u, <math>

<math> Y(u,v) = r \, (1 + \cos v) \, \sin u, <math>

<math> Z(u,v) = - \hbox{tanh} \left( {2 \over 3} (u - \pi) \right) \, r \, \sin v,<math>

where both u and v range from 0 to 2π. These equations are similar to those of a torus. Figure 1 shows a closed cross-cap.

Figure 1. Two views of a cross-cap.

A cross-cap has a plane of symmetry which passes through its line segment of double points. In Figure 1 the cross-cap is seen from above its plane of symmetry z = 0, but it would look the same if seen from below.

A cross-cap can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points. The result is shown in Figure 2.

Figure 2. Two views of a cross-cap which has been sliced open.

Once this exception is made, it will be seen that the sliced cross-cap is homeomorphic to a self-intersecting disk, as shown in Figure 3.

Figure 3. Two alternate views of a self-intersecting disk.

The self-intersecting disk is homeomorphic to an ordinary disk. The parametric equations of the self-intersecting disk are:

<math> X(u,v) = r \, v \, \cos 2 u, <math>

<math> Y(u,v) = r \, v \, \sin 2 u, <math>

<math> Z(u,v) = r \, v \, \cos u, <math>

where u ranges from 0 to 2π and v ranges from 0 to 1.

Projecting the self-intersecting disk onto the plane of symmetry (z = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself).

The plane z = 0 cuts the self-intersecting disk into a pair of disks which are mirror reflections of each other. The disks have centers at the origin.

Now consider the rims of the disks (with v = 1). The points on the rim of the self-intersecting disk come in pairs which are reflections of each other with respect to the plane z = 0.

A cross-cap is formed by identifying these pairs of points, making them equivalent to each other. This means that a point with parameters (u,1) and coordinates <math> (r \, \cos 2 u, r \, \sin 2 u, r \, \cos u) <math> is identified with the point (u + π,1) whose coordinates are <math> (r \, \cos 2 u, r \, \sin 2 u, - r \, \cos u) <math>. But this means that pairs of opposite points on the rim of the (equivalent) ordinary disk are identified with each other; this is how a real projective plane is formed out of a disk. Therefore the surface shown in Figure 1 (cross-cap with disk) is topologically equivalent to the real projective plane, RP².

See also: Boy's surface, Roman surface.