

When browsing another thread, I saw a claim that hexagonal grid models natural continuous movement more accurately than the square grid. I have seen this claim also before, so I decided to make a thread for it.
I cannot see why hexagonal grid would be a better model than square grid. In hexagonal grid you can move only in six directions, whereas in a square grid you can move to four or eight directions depending on whether you allow diagonal movement. Four, six and eight are all small finite numbers, and are all practically equally far from the infinity of directions in the natural world.
My hypothesis is that the square grid is easier to grasp than the hexagonal grid, and hence exploiting the shortcomings of the square grid is easier than on a hexagonal grid. So I suppose because of that hexagonal grid FEELS a better model for continuous movement, but in reality there's no difference.


Ralph T
United States Signal Hill California

But do the math on diagonal movement. It's 1.4 (square root of 2 per Pythagorean theorem) times the distance of orthogonal movement. If you let a player move diagonally across a square they should be charged 1.4 times movement instead of 1. In contrast hexagonal movement is the same distance whatever direction you go.




As said by the previous poster the difference is huge.
If you have four directions units who want to move one space diagonally must move two spaces. That's not desirable.
If you have eight directions units moving two diagonal spaces has the same result as moving two spaces foward. Which means units moving diagonally somehow move faster.




General_Norris wrote: As said by the previous poster the difference is huge.
If you have four directions units who want to move one space diagonally must move two spaces. That's not desirable.
On hexagonal grid there are only six privileged directions. If you want to go in a direction that's not one of them, you have to waste movement just like described in the quote.




ralpher wrote: But do the math on diagonal movement. It's 1.4 (square root of 2 per Pythagorean theorem) times the distance of orthogonal movement. If you let a player move diagonally across a square they should be charged 1.4 times movement instead of 1. In contrast hexagonal movement is the same distance whatever direction you go.
Unless you wish to go in the direction of one of the corners of your hex  then you are charged 2 move to travel only 1.5 times the centrecentre distance of two adjacent hexes. Now, 2/1.5 ~1.3, which is a little better than the 1.4 discrepancy on a square grid.
Me, I just like hex maps for no rational reason.
Edit: This is wrong, I have been corrected below


todd mewborn
United States toledo Ohio

There are some games (I just can't think of the game right now) that use hexes and have 12 point facing. You would move from hex to hex spine.


Rich Shipley
United States Baltimore Maryland
the liberal unsavory type

tmewborn wrote: There are some games (I just can't think of the game right now) that use hexes and have 12 point facing. You would move from hex to hex spine.
I've got a miniatures game in my head that works like this.




old_gamer wrote: Unless you wish to go in the direction of one of the corners of your hex  then you are charged 2 move to travel only 1.5 times the centrecentre distance of two adjacent hexes. Now, 2/1.5 ~1.3, which is a little better than the 1.4 discrepancy on a square grid.
Actually, the relation is 2 / sqrt(3) then, which is around 1.15.
To the OP  hexagonal tiling allows for better representation of movement when simplicity of movement rules is to be maintained and movement range of a unit is not large in terms of number of steps (squares, hexes) that it can cover. If those requirements do not apply, you can use square grid and make orthogonal move cost 1 point and diagonal cost 1.5 points (or 2 and 3 points, respectively), for relative simplicity, but if different fields represent various terrain types with varying entering costs and orthogonality is not very important to the modelled space (like terrain in a wargame, unlike playing field in a sports game), it is easier and less cumbersome to represent terrain with hexagonal grid.


Mike StClair
United States San José California

I remember in D&D 3rd edition they attempted to solve this by having diagonal movement cost 1.5 movement  so each diagonal movement would alternate between costing 1 and 2 movement points. It was a clunky way to handle it, but it made more sense than having diagonals essentially be a free movement.
Either way I prefer hexes, and I use them by default if the game can be easily adapted to it. It doesn't always mesh perfectly with nonnatural environments (cities and buildings) but it's still better, in my opinion.


Ted Groth
United States Appleton Wisconsin

old_gamer wrote: ralpher wrote: But do the math on diagonal movement. It's 1.4 (square root of 2 per Pythagorean theorem) times the distance of orthogonal movement. If you let a player move diagonally across a square they should be charged 1.4 times movement instead of 1. In contrast hexagonal movement is the same distance whatever direction you go. Unless you wish to go in the direction of one of the corners of your hex  then you are charged 2 move to travel only 1.5 times the centrecentre distance of two adjacent hexes. Now, 2/1.5 ~1.3, which is a little better than the 1.4 discrepancy on a square grid. Me, I just like hex maps for no rational reason.
Actually, if the desired movement is in the direction of the corner of the hex, then you are charged 2 moves to travel 1.732 times the center to center distance of the hexes (not 1.5) for a 13.4% loss of efficiency. The diagonal move on a square grid would charge 2 perpendicular move to travel 1.414 times the center to center distance of the squares. for a 29.3% loss of efficiency.
But a more fair comparison would be to consider movement of approximately two spaces, in any of 12 facing points in a square grid vs. a hex grid: The hex grid has already been described above, 2 points charged to move 2 centercenter distances through the hex sides (60 degree facings) or 2 points charged to move 1.732 centercenter distances through the hex corners (30 degree facings). The closest equivalent on a square grid would be to move out 2, and then perpendicularly 1 space. This would be a facing of about 26.6 degrees, and would charge 3 points for an effective distance of 2.236 centercenter distances, which is still a 25.5% loss of efficiency, almost as bad as above.
So a hex grid clearly handles off axis movements more effectively.
But hex grids create problems with facings of defensive lines of units. If orientation of the defensive line is parallel to the sides of the hexes, then every other unit is forced out slighly ahead of the line, exposing these units to potential attack on three sides, while the intervening units are exposed on only one side. Three adjacent attackers can concentrate on a single defender, and create a break in the defensive line. (the other line orientation on a hex grid exposes each and every defender on two sides) The square grid on the other hand exposes only one side of each defender when the defensive line is parallel to the sides of the square, or each defender is exposed on two sides if the defensive line is on the diagonal.


Gregorio Morales
Spain Alicante

ralpher wrote: It's 1.4 (square root of 2 per Pythagorean theorem)
You don't really need Pythagorean Theorem to show that it's sqrt(2)


Scott Hill
United Kingdom Cambridge Cambridgeshire

Maybe hex grids should be equilateral triangle grids instead, and you'd move from intersection point to intersection point, rather than hex centre to hex centre, then all movement, in any one of 6 directions, would cost 1 irrespective of direction, and there would be no diagonals.


Ted Groth
United States Appleton Wisconsin

Railoc wrote: Either way I prefer hexes, and I use them by default if the game can be easily adapted to it. It doesn't always mesh perfectly with nonnatural environments (cities and buildings) but it's still better, in my opinion. Remember you can use an offset square grid (brick pattern) to approximate a hex grid, and better match the 90 degree angles found in building structures.edit: It can create some dead halfspaces along the walls though.
or use offset rectangles 1 unit x 0.866 units, to exactly duplicate the hex grid distances. This allows you to intermingle hexes and offset rectangle bricks on the same map, if you use a little ingenuity in the artwork. edited to add images


Ted Groth
United States Appleton Wisconsin

Scorpion0x17 wrote: Maybe hex grids should be equilateral triangle grids instead, and you'd move from intersection point to intersection point, rather than hex centre to hex centre, then all movement, in any one of 6 directions, would cost 1 irrespective of direction, and there would be no diagonals. The result is exactly the same, it just looks different.




Scorpion0x17 wrote: Maybe hex grids should be equilateral triangle grids instead, and you'd move from intersection point to intersection point, rather than hex centre to hex centre, then all movement, in any one of 6 directions, would cost 1 irrespective of direction, and there would be no diagonals. It is equivalent to hexagonal grid, the same movement overhead in unprivillaged direction.


Scott Hill
United Kingdom Cambridge Cambridgeshire

Tradewinds Ted wrote: Scorpion0x17 wrote: Maybe hex grids should be equilateral triangle grids instead, and you'd move from intersection point to intersection point, rather than hex centre to hex centre, then all movement, in any one of 6 directions, would cost 1 irrespective of direction, and there would be no diagonals. The result is exactly the same, it just looks different.
I know.




old_gamer wrote: Unless you wish to go in the direction of one of the corners of your hex  then you are charged 2 move to travel only 1.5 times the centrecentre distance of two adjacent hexes. Now, 2/1.5 ~1.3, which is a little better than the 1.4 discrepancy on a square grid. Me, I just like hex maps for no rational reason.
aplegat2 wrote: Actually, the relation is 2 / sqrt(3) then, which is around 1.15.
Tradewinds Ted wrote: Actually, if the desired movement is in the direction of the corner of the hex, then you are charged 2 moves to travel 1.732 times the center to center distance of the hexes (not 1.5) for a 13.4% loss of efficiency. The diagonal move on a square grid would charge 2 perpendicular move to travel 1.414 times the center to center distance of the squares. for a 29.3% loss of efficiency.
But a more fair comparison would be to consider movement of approximately two spaces, in any of 12 facing points in a square grid vs. a hex grid: The hex grid has already been described above, 2 points charged to move 2 centercenter distances through the hex sides (60 degree facings) or 2 points charged to move 1.732 centercenter distances through the hex corners (30 degree facings). The closest equivalent on a square grid would be to move out 2, and then perpendicularly 1 space. This would be a facing of about 26.6 degrees, and would charge 3 points for an effective distance of 2.236 centercenter distances, which is still a 25.5% loss of efficiency, almost as bad as above.
So a hex grid clearly handles off axis movements more effectively.
Yep, both right.
You know, I was writing that thinking "Am I sure I have this right? I've been having a sleepy/dopey day. Nah, how could anyone get this wrong?"
Thanks for the correction, both of you.


Ted Groth
United States Appleton Wisconsin

Tradewinds Ted wrote: Actually, if the desired movement is in the direction of the corner of the hex, then you are charged 2 moves to travel 1.732 times the center to center distance of the hexes (not 1.5) for a 13.4% loss of efficiency. The diagonal move on a square grid would charge 2 perpendicular move to travel 1.414 times the center to center distance of the squares. for a 29.3% loss of efficiency. Thinking again about the square grid, if the rules are set so the diagonal only costs the same 1 point, then there is a 41.4 gain in efficiency. But if the diagonal is set a a cost of 1.5 points, then there is only 5.7% loss of efficiency instead. (but charging fractional movement points can prove clunky )
Quote: But a more fair comparison would be to consider movement of approximately two spaces, in any of 12 facing points in a square grid vs. a hex grid: The hex grid has already been described above, 2 points charged to move 2 centercenter distances through the hex sides (60 degree facings) or 2 points charged to move 1.732 centercenter distances through the hex corners (30 degree facings). The closest equivalent on a square grid would be to move out 2, and then perpendicularly 1 space. This would be a facing of about 26.6 degrees, and would charge 3 points for an effective distance of 2.236 centercenter distances, which is still a 25.5% loss of efficiency, almost as bad as above. again if the diagonal costs only 1 point, then this would be a charge of 2 for a movement of 2.236, or an 11.8% increase in efficiency. If the diagonal costs 1.5 the this would be a charge of 2.5 for the 2.236 distance, for a 10.6% loss of efficiency
Quote: So a hex grid clearly handles off axis movements more effectively. This is still true if the diagonal costs only 1 point. If the cost of diagonal movement is set at 1.5, then the efficiency of off axis movement is actually a bit better on the square than the hex grid, at least for moderate movements. But the fractional movement cost can be a problem for writing clean rules.


Roger McKay
Canada Bedford Nova Scotia

tmewborn wrote: There are some games (I just can't think of the game right now) that use hexes and have 12 point facing. You would move from hex to hex spine.
SPI's Air War college course had that.
Somehow, I think at least one version of Star Fleet Battles had it. Perhaps as an optional rule?
I have been war gaming since the mid'70s, so hex maps just feel right to me, but they don't handle orthogonal buildings very well. So, I prefer the square grid for RPGs.



