There are three regular tilings on the plane - square, triangular, and
hexagonal. If you ignore the tiling, and just look at where the center
of each tile is, you get the following three patterns:

Some board games let you place pieces in one of these patterns:
checkers, chess, and Go use the first, while hex and chinese checkers
use the second. I don’t know of any game that uses the third, other
than unusual Go and hex variants.

In the US, we are used to placing pieces at the center of tiles. But
Go uses a different convention - you place pieces at the intersections
of the lines. Go still uses a square grid; you just place the pieces
differently. But you can’t put the points from the centers of a
hexagonal grid onto the vertices of a hexagonal grid; they make
different patterns. Instead, you have to put them on a triangular
grid. Likewise, points from the centers of a triangular grid may be
placed on the vertices of a hexagonal grid. There’s a name for this:
triangular and hexagonal tilings are
duals of each other, and a
square grid is its own dual.

I’m interested in the game of Hex. It uses the second pattern of
points, which means it can be drawn on either a triangular or
hexagonal grid. It is traditionally drawn on a hexagonal grid, though
I find the triangular grid easier on the eyes. There is a surprising
lack of triangular hex board grids on the internet, though, so I wrote
one in Postscript. I wanted it to fit Go stones and be at least size
11, making it too big to fit on one page. So here is half of a hex
board