Over at Fractal Forums, DarkBeam came up with a Distance Estimator for a trefoil knot in this thread. Here are a few samples, created using the new ‘soft’ raytracer, I’m working on in Fragmentarium:

These kinds of knots are easy to describe by a parametrized curve, but making a distance estimator for them is impressive – I wouldn’t have guessed it was possible at all.

It is also possible to create several variations:

In the same thread, Knighty came up with an impressive figure-8 knot distance estimator:

In another thread at Fractal Forums, Knighty also published an interesting technique (“Fold and Cuts”) for creating a large variety of distance estimated polyhedra:

(If you wonder about the materials, I’ve added some 3D Perlin noise to the distance estimate – this is simple way to creature a structural texture, and it creates true displacements, not just surface normal perturbations).

The threads linked to above contains Fragmentarium scripts with the relevant distance estimators.