The Trefoil Knot

In mathematics, even the simplest things can have an astounding depth. Let’s for instance take the trefoil knot, the simplest knot there is:

Trefoiltube

One can replace the tube by a ribbon, like so:

Trefoil2

This could be done with a simple ruled surface, but I like a challenge. To make this a minimal surface, one can use Björling’s formula. The game becomes tricky if one wants the surface to be of finite total curvature, but this can be done as well. Then it is not difficult to let the normal of the surface rotate once to get a knotted minimal Möbius strip.

Trefoilmobius

Faster spinning normals create knotted helicoids.

Trefoil30

Extending the surface beyond a small neighborhood of the trefoil knot makes things appear really complicated.

Trefoilbig

Of course the same can be done with more complicated knots.

Quatrefoil

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