[seqfan] Re: Knight's tour on infinite board

[M. F. Hasler]

On Sat, May 4, 2019, 20:50 Hans Havermann <gladhobo at bell.net> wrote:

HP: "Combining all spiral variants with the enumeration of various paths on
the 2-d lattice or space filling curves like the knight's tour would be a
can of worms."

I've actually thought about it a bit. There appear to be 8 square spiral
variants: First step east, north, west, or south, followed by either a
clockwise or counterclockwise numbering(.…)


These tours could also be coded using "relative" co-ordinates rather than
absolute (x,y) coordinates.
A very natural encoding could be to denote by 0, ±1, ±2..., respectively, a
move in the same direction as the preceding one, or involving a "turn" by
that multiple of 360°/8  to the left or right, respectively.

That way the encoding of the Knight tour becomes independent of the
starting direction  (and the encoding more compact). Rotational invariance
is intrinsic and invariance wrt reflections amounts to changing the sign of
all terms.

- Maximilian