John Conway has a very interesting theory of "tangles" where he associates a continued fraction to a wide class of kntots http://www.maths.ed.ac.uk/~aar/papers

John Conway has a very interesting theory of "tangles" where he associates a continued fraction to a wide class of kntots http://www.maths.ed.ac.uk/~aar/papers/conway.pdf http://www.maths.ed.ac.uk/~aar/papers/conway.pdf There is also the Conway Tesselation here he sub-divides a triangle into 5 equal pieces, and creates a non-periodic tiling this way https://en.wikipedia.org/wiki/Pinwheel_tiling https://en.wikipedia.org/wiki/Pinwheel_tiling