Self-avoiding walks: other grids

The previous blog post showed some examples of self-avoiding paths, all on orthogonal (Cartesian) grids. But there are lots of other grids. Two other “natural” grids are hexagonal and triangular.

The hexagonal grid looks like beeswax, or chicken wire (known as “gopher wire” in parts of the world where gophers are troublesome). That is, the edges of the grid are the edges of hexagons in their standard packing.

The triangular grid is just what it says: a grid of triangles. One way to create such a grid is to place lines \(y-x=n\) for all integer \(n\) on the cartesian grid. This grid can be adjusted slightly by an affine transform to be a grid of equilateral triangles.

Here’s an example:

There would be a lot of these, given the number of possible choices at each vertex.