Surfaces from the Möbius Strip to the K3 Surface

Start with a strip of paper whose ends are joined with a half-twist. Möbius and Listing identified it independently in 1858. It has a single boundary curve and only one side, so an object sliding along it returns as a mirror image of itself, and no consistent clockwise direction can be defined across it.

Felix Klein described this surface in 1882. Like the Möbius strip it is one-sided and non-orientable, but it is closed, with no boundary at all. It cannot sit in three-dimensional space without passing through itself, though it embeds cleanly in four dimensions.

Werner Boy found this surface in 1901 after his advisor David Hilbert asked him to prove that the real projective plane could not be immersed in three-dimensional space. He produced an immersion instead. Bernard Morin gave the first explicit parametrization in 1978, so the surface was drawn in full only decades after its discovery.

The torus returns to the familiar, two-sided case. It is made by revolving a circle around an axis in its own plane that does not pass through it, giving the doughnut shape of a ring torus. Topologically it is the product of two circles, a compact surface of genus one. When the radii change it becomes a horn torus or a self-intersecting spindle torus.

A Seifert surface is a connected, orientable surface whose boundary is exactly a given knot or link. In 1934 Herbert Seifert gave an algorithm for building one from any knot diagram. The smallest genus among all such surfaces for a knot is an invariant of the knot itself, so studying a surface becomes a way of studying a curve.

J. W. Alexander discovered this wild embedding of a sphere in 1924. With its interior it forms a topological ball in which every loop can be shrunk to a point, yet its exterior is not simply connected. It shows that the Schoenflies theorem, true for closed curves in the plane, fails in three dimensions.

Here the sphere takes on a complex structure. The Riemann sphere is the complex plane plus one point at infinity, arising from Bernhard Riemann's work on complex analysis in the 1850s. Stereographic projection wraps the whole plane onto the sphere, so every point of the plane and the point at infinity match exactly one point of the surface.

A Riemann surface is a connected one-dimensional complex manifold. Near every point it looks like a patch of the complex plane, but globally it may resemble a sphere, a torus or several sheets glued together, as with the graphs of the square root or the logarithm. A two-dimensional real surface gains such a structure only if it is orientable, which is why the Möbius strip and Klein bottle from the start of this trail are excluded.

An elliptic curve is a smooth projective algebraic curve of genus one with a chosen point that serves as the identity of a group law. Over the complex numbers these curves correspond to embeddings of the torus, which links this stop back to the fourth one. They figured in Andrew Wiles's proof of Fermat's Last Theorem and are used in elliptic curve cryptography.

A K3 surface is a compact connected complex manifold of dimension two with a trivial canonical bundle and irregularity zero. André Weil gave these surfaces their name in 1958 and made influential conjectures about their classification. They are among the simplest algebraic varieties that do not reduce to curves, and they have been applied to mirror symmetry and string theory.