A Beautiful Mess!

Oops!

Turn a square grid on it's side diagonally from pole to pole and stretch sideways around the circumference of the world. The now diamond shapes subdivide into triangles, although two at the poles is a problem, until we subdivide each triangle six ways so we have four around each pole. If we have so many thousand kilometer diamond quads from pole to pole, there would (rows-1)^2 'hexagons' plus rows times four edge bits, making (rows-1)*2 half hexagon pairs wrapping east to west, two polar quads plus one on the equator, which simplifies to rows^2+1, although the equator bit would be happier as part of it's northern and southern neighbors and the polar quads are also surrounded... I ended up starting a novel there!
All I really need is just enough subdivision to not have edge-case weirdness at the grid corners and random walk through the vertices define land vs sea.