ochoin  06 Feb 2016 3:39 a.m. PST 
link Interesting but I have better uses for the 'hard earned'. 
Last Hussar  06 Feb 2016 3:46 a.m. PST 
Caveat – Certain kinds of negative comment will be ridiculed. Anyway… You are going to grid a game, rather than use free movement. Which kind of grid do you use 1) Traditional 'squares' (as in a chessboard) – Easy to make but with the 'Diagonals problem' 2) Hexes – Purist, but difficult to grid an meaningful area easily in. 3) Offset rectangle (like brickwork pattern) – an easy draw version of hexes which reduces the diagonal distortion (I think a 5:4 ratio works, might be 10:7). Not actually seen this done. 4) Triangles (The only other shape that tessalates – here for No Damn Good Reason) 5) Area – Irregular areas, those that are easy to traverse are larger than slower going. 
Martin Rapier  06 Feb 2016 4:02 a.m. PST 
I find squares work well for more linear warfare (so really, Ancients right up to late nineteenth century) , whereas hexes are better for more dispersed/free wheeling periods (1900 onwards). The main exception is set piece assaults in high intensity warfare (think WW1, Kursk etc) where squares are a better model of unit boundaries and axes. Never really found diagonals a problem. It is easiest to make all movement orthogonal, and borrow the slick mathematical trick from PBI for measuring ranges (all ranges are orthongal but you are allowed one diagonal – brilliant, simple and a reasonable approximation). 
Martin Rapier  06 Feb 2016 4:11 a.m. PST 
Sorry, meant to add, I generally prefer area grids (think, Lost Battles, Square Bashing, PBI etc) to one unit per grid point. The latter is far more fiddly to manage and can lead to over complication, but it depends on the scope of your simulation. When I converted all the Neil Thomas rules to use squares or hexes, it was easier to go with one unit per node (apart from some stacking for single base units), whereas for e.g. my hex based version of Spearhead it was far easier to allow varying numbers of elements per hex as modern units spread out or clump up and don't operate in fixed close order formations. 
etotheipi  06 Feb 2016 4:36 a.m. PST 
Penrose Tiles – They give a nice breakup of the area into equal, but nonrepeating patterns giving ground actual irregularity without doing anything but counting spaces. NonRegular Hexagons – Saw a paper about at a military simulation conference a few years ago. If you align two opposite points of a hexagon to a quarter grid based on the width of the parallel sides, you get hexes that are easy to lay out, and give you closer angle distances than squexes. Equilateral Triangles – Good for whole globe planetary games, and they make a recursive grid. There are an infinite number of shapes that tessellate. There are only two (or three) regular shapes that tessellate – triangles and squares (and hexes, unless you count them as composed triangles). 
Andy Skinner  06 Feb 2016 4:54 a.m. PST 
Diagonals on a square grid can also be handled with "every other diagonal step costs 2", or "orthogonal costs 2 and diagonal costs 3". etothepi, got an illustration for the nonregular hexagons? I don't find the offset squares help for looking along the axes. andy 
Bob the Temple Builder  06 Feb 2016 4:58 a.m. PST 
As I use grids almost to the exclusion of anything else, I support Martin Rapier's comments 100%. 
Who asked this joker  06 Feb 2016 5:04 a.m. PST 
I too like Matins comments. I will point out that Hexes can work pretty well for linear warfare if the hex spines are oriented correctly. If all units have to point forward to a hex spine (vertex) then the line is perfectly straight. Of course a move forward is either forward and to the left or forward and to the right. 
(Phil Dutre)  06 Feb 2016 5:48 a.m. PST 
You can find a lot of different tilings. The topic has also been studied extensively in mathematics: regular tilings, periodic, aperidioc, convex, rotationalsymmetric, etc. a couple of years ago I published a paper about Wangtiles (i.e. Tiles with colored edges), and their aperiodic properties as a side project. Anyway, squares, hexes, and offset squares seem to do the job for gaming purposes. Hexes and offset squares are largely identical, but there are subtle differences such as lines connecting centres that pass through different cells. This might or might not be an issue in the ruleset at hand. As for squares, I second the solution of one diagonal move allowed during movement. It is an almost exact approximation to a discretized circle, especially for a low number of squares. 
Martin Rapier  06 Feb 2016 10:45 a.m. PST 
On a purely practical note, for large area movement systems (like Lost Battles with a 5x4 grid) don't even need a special cloth or terrain system. You can just mark the corners of the grid with bits of terrain on a normal cloth, which also looks rather nice. 
Mako11  06 Feb 2016 12:34 p.m. PST 
As probably mentioned above, you can use both squares and hexes for nonregular, as well as regular movement too. Squares can be moved across at 45 degree angles, and you can move across hexes in 60, or 30 degree angles as well. For the latter, you just need to think outside the box, and not require your figure bases to align with the flats of the hexes. They can also be aligned with the points of them, and move across them too. Both work quite well. For the squares, drawing in the 45 degree gridding might help, but shouldn't be absolutely necessary. 
Ottoathome  06 Feb 2016 2:36 p.m. PST 
I used a regular square grid for a decade when I was playtesting rules. There is no "problem" with the diagonal. You just count squares not space. If everything is measured by the grid, then a square is a square is a square, even if the actual distance in inches is greater on the diagonal. If you're using a grid THE USE THE GRID! I presently have a grid on my table top. It's hexagonal with 12" between the parallel sides of the hexagon. But this is only for terrain. I model my terrain on these large hexes to make it more rugged and easily modular. The movement is free movement nor is firing or anything done on the grid. The grid DOES delineate terrain types within it, but that's it. Thus the grid will be open, road, rough, very rough, block, and urban. The terrain within is undifferentiated as far as stone walls, buildings etc. Remember I deal in big battles in he 18th century so the idlyfiddly methods most gamers wants simply doesn't apply. 
Zephyr1  06 Feb 2016 2:53 p.m. PST 

etotheipi  06 Feb 2016 3:45 p.m. PST 
etothepi, got an illustration for the nonregular hexagons?
The proceedings of the conference are here. Search for "hexagon"; the paper is toward the end. Lots of diagrams. And data on the delta between the irregular ones proposed and regular ones. Seems to be a reasonable compromise between ease and accuracy. 
RudyNelson  08 Feb 2016 4:23 p.m. PST 
Area is not often used for tactical games. However area and point to point are preferred in campaign system when bringing armies into battle fast is desired. Prerailroad warfare also favors these two systems. For a tactical system, it really depends on the designer's concept. I have been around so long that I have seen them all used and the prehex concept by Zocchi called the brick system which was used in his Battle of Britain game. It can be used for tactical as well. Most seem to prefer the hex as it gives you six clear movement directions. The square give four clear and if diagonal are allowed you have 8 choices. Some seminars have compared squares as tactical araes in concepts. 
N Drury  09 Feb 2016 6:10 a.m. PST 
For a brick pattern to closely approximate a hex grid, i.e. the centres of the six 'bricks' surrounding a given point are equidistant, a ratio of length:side of 15:13 is very close. 