Hexes have a cartographic advantage over grids in that the center of a hex is equidistant from the centers all six adjacent hexes. In contrast, on a standard graph paper grid diagonal movement is more efficient than moving in a cardinal direction, assuming a destination other than cardinal-adjacent (that is, other than due north, due south, due east, or due west).
Recently I noticed that squares in a brick configuration are topologically similar to hexes in terms of adjacency. Each brick is adjacent to six surrounding bricks.
To see another way how bricks are similar to hexes, consider the following image and imagine the orange brick overlay moving right until the center of the bricks is superimposed over the center of the hexes.
(This post is groundwork for another idea. To be continued!)