u/hilberts_drinking_pr

I would like to share a few images of spiral knight tilings inspired by this video from Numberfile. Most were sampled from this page. Hope you find these fun!

Setup: we number the squares of the infinite chessboard in a spiral order, starting at the origin and walking outward counterclockwise. We place chess knights one at a time, cycling through n colors. Each new knight goes in the earliest spiral cell that is still empty and satisfies a constraint determined by a directed graph on the n colors:

arc u -> v means "a u-knight at a knight-move from cell c forbids placing color v at c."

Other examples and a small illustration of how these are constructed can be found in my earlier post.

Apologies if I'm beating a dead horse, but I've had a lot of fun experimenting with knight tiling over the last few days. With Claude's help, I set up a little UI to generate images like the ones above, in case anyone is is interested in seeing more: https://yakymp.github.io/spiral-knight-tiling/

The Numberfile video that prompted this exploration can be found here and a previous post with more background information can be found here.

Here are a few more knight tilings on a square spiral, based on this awesome Numberfile video.

We number the squares of the infinite chessboard in a spiral order, starting at the origin and walking outward counterclockwise. We place chess knights one at a time, cycling through n colors. Each new knight goes in the earliest spiral cell that is still empty and satisfies a constraint determined by a directed graph on the n colors:

arc u -> v means "a u-knight at a knight-move from cell c forbids placing color v at c."

The legend at the bottom shows the underlying graph and the early development of the grid (number in a cell indicates the turn on which it was placed).

I just watched the new Numberfile video and thought it would be fun to generate similar plots based on Tourney graphs. For example, if G = {red -> black}, we can place a red knight in any open cell and a black knight in any open cell not attacked by an existing red knight. Here are some examples.