[rrobukef]

On the other hand, for movement along the 8 directions the distance is 41% short, for movement along the 4 main directions, the taxicab distance is 41% longer than the actual distance.

For a hexagonal grid, the distance is approximated by counting the number of cells of the shortest path. This has <10% relative error (IIRC) from the actual distance.

[alisonkisk]

If you take diagonals to be 1.5, then distances are <10% high.

The half-short-side heuristic (sqrt(aa + bb) ~= a + b/2) makes taxicab very close to correct for non-square distance as well.

[rrobukef]

But only if you move multiple cells in one turn. If movement is restricted to one cell each turn, your field distorts.

The same happens when you split your movements: For example: player A moves 6 each turn; player B moves 2 thrice in a turn. Conclusion: A outruns player B at speed (4,4)/turn vs (3,3)/turn. (For A d(4,4)=6; B has d(1,1)=1.5)

[aidenn0]

Yes, hex grids are better when the number of cells moved per turn is very small.

Handling multiple moves in a single turn is easy either by doubling the movement points you get (then N/S/E/W is 2 points, diagonal is 3), or informally by treating every other diagonal move as costing 2 points.

[aidenn0]

Diagonal moves equal to horizontal/vertical moves is horrible. I don't remember which D&D suggested that, but "circles" being square is just terrible.