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I've used the corners of a square-gridded board to increase the density
of locations on a gameboard - you get 2x the spots to move onto for no
physical increase in size. As far as basic geometry, all this does is
shrink the squares and rotate them 45 degrees. The connectivity is the
same. It works like this: a square has 4 'nearest' neighbors, the 4
orthogonally adjacent squares. But a square also touches another 4
diagonally, so it has 8 touching neighbors. The diagonal neighbors are
farther away than the orthogonal ones by a factor of 1.4... but there are
still 8 neighboring/touching locations, a 2 different distances, 1 & 1.4.

Analogously, we look at hex boards and see 6 nearest neighbors [distance
of 1], and no others touching. We see 12 next nearest neighbors, all at a
distance of 2. But we see that some hexes [6] are orthogonally and
linearly connected to the center hex, but others [6] are not. They are connected in a zig-zag pattern of hex center to hex center, and a straight line passes between other hexes, along hexsides. This is what we consider a 'hex diagonal', and it works analogously to 'square diagonals'. 

Look at this board: http://chessvariants.wikidot.com/universal-board
This uses the octagonal-square tiling. An octagon has 8 nearest neighbors,
and a square has 4. If you use figure centers and intersections, the
octagon has 16, with the 8 intersections becoming the 'nearest' [in
physical, measured distance between figure (geometric) centers and
intersections], and the 8 centers now representing 'diagonals'. The
square here gets 8, its 4 corners as the 4 nearest, and the 4 octagons it shares an edge with. If you look at the square-octagon 'edge line diagonal', analogous to the hex 'diagonal', you see there are 4 squares at the ends of the 4 lines radiating from the corner of any square. 

Now look at the octagons, using the octagon-square 'edge-diagonal'. Each
octagon has only 4 edge-diagonal neighbors to go along with its 8
'orthogonal' neighbors, for a total of 12 possible moves, if I'm figuring
this right. [Heh, I'm not sure I want to read what I just wrote!] Anyway,
that's only 3/4 of the 16 you get from using figure centers and vertices,
so this pattern doesn't transform. [Lol, Ben, rescue me here...]