The Properties of Hexes and Mapping

    One of the most common tools people will come across in tabletop RPGs are hex maps. They divide a map into a number of hexagonal sections to ease measuring distances, plotting trips, and filling in new areas. Obviously, this comes with limitations that a plain map does not have and can sometimes pull people out of the fiction that they play in, but hex maps have their uses, even if only for the DM.

    The problem with hex maps is that it is normally important to have several maps at different scales, such as one for nearby areas, one for the kingdom, and maybe a world map. Obviously all of these levels need to agree with each-other, but what scales to put them at is a pertinent question.

    Unfortunately, it is one that does not have any single satisfactory answer. Still, I will be working to discover what I believe are the best sets of mapping scales for use in the general OSR space. For those not interested in reading the article and my reasoning, here are what I consider the two best sets of scales for hex-mapping:

  • 220 yard (x8)> 1 mile (x5)> 5 mile (x5)> 25 mile
  • 264 yard (x10)> 1.5 mile (0.5 League) (x4)> 6 mile (x4)> 24 mile

Existing Guidance

    Obviously the first place to look for guidance on these matters are rulebooks. For brevity, I will only go through a couple.

OD&D

    The original version of D&D certainly relied heavily on Outdoor Survival for its maps, but it still provided some guidance. It gives travel times in terms of hexes, with the assumption that each hex is at most "about 5 miles" across. While this could be interpreted to mean that one should use 5 mile hexes, it could also mean that the long diagonal (corner to corner) of the hex should be 5 miles. In this case, you should use hexes that are between 4 and 5 miles across normally. There are no mentions of different scales that I saw.

AD&D

    The dungeon-master's guide includes mentions of a 200 yard hex when clearing terrain for a base, as well as 1-mile hexes, and a 30-mile campaign hex in the same section. Elsewhere, it claims that the campaign hex should be anywhere from 20 to 40 miles per hex and that these campaign hexes should probably fit 5 smaller hexes on a face (effectively 7 or 8 hexes across depending on how you place it), which would imply a map of hexes ranging from 4 to 8 miles across as well.

ACKS

    Of the OSR games, ACKS has a particular focus on kingdom and domain level play, which has given it a very detailed set of directions for mapping and making use of hex maps. It suggests the usage of two levels of maps: a regional map using 6 mile hexes and a campaign map using 24 mile hexes.

Hex Sizes

    A hex's size (in RPG purposes) is defined as the distance one must go to get from the center of one hex to an adjacent hex. Coincidentally, this is equal to the distance of a hex's short diagonal (edge to edge). The other important distances on a hex are the long diagonal (corner to corner) and the face length. Because of how the math works, assuming we start from the short diagonal, the face is that divided by the square root of 3 and the long diagonal is double that.

    Obviously, this gives some unfortunate numbers if you aren't proceeding in one of the six hex directions. However, there are a few hex sizes which give some nice numbers if you are willing to overlook a small bit of rounding:

  • 6 miles - 3.5 mile face, 7 mile diagonal
  • 7 miles - 4 mile face, 8 mile diagonal
  • 13 miles - 7.5 mile face, 15 mile diagonal
  • 19 miles - 11 mile face, 22 mile diagonal
  • 20 miles - 11.5 mile face, 23 mile diagonal
  • 26 miles - 15 mile face, 30 mile diagonal
  • 32 miles - 18.5 mile face, 37 mile diagonal
  • 39 miles - 22.5 mile face, 45 mile diagonal
  • 40 miles - 23 mile face, 46 mile diagonal
  • 45 miles - 26 mile face, 52 mile diagonal
    If you are willing to be extra generous with your rounding, a 5 mile hex has a 3 mile face and a 6 mile diagonal as well. Still, based on this it is apparent that these "ideal math hex sizes" are less than ideal for mapping in most cases. After all, who would even want to attempt working with a 13 or 19 mile hex map? Crazy people, that's who.

    In any case, the ideal hex size does appear to be 5, 6, or 7 mile hexes. 6 has the disadvantage of including a .5 in its math, but 7 is a particularly un-round number and 5 requires generous rounding, so it should be a matter of taste which you prefer.

Tiling Hexes

    When it comes to tiling hexes, that is having hexes come together to form a larger hex, some sizes tile better than others. Through my experiments, there are two types of good tiling: ideal and semi-perfect. Ideal tiling has a super hex that uses the corners of existing hexes in a neat way and has a single central tile to make zooming particularly easy. Semi-perfect tiling also uses corners, but cuts them instead of following them while still having a single central tile. Basically, the difference is at the tile where the super-hexes meet. Ideal is where they follow the existing tile lines. Semi-perfect is where they ignore the existing tile lines.

    So, based on this, I have calculated the sizes. Ideal tiling begins at 4 hexes across and increases by 3 each time (4, 7, 10, etc.). Semi-perfect tiling begins at 5 hexes across and increases by 3 as well (5, 8, 11, etc). No matter which is chosen, there should not be mixing between the two types due to the different patterns that go into mapping them.

Potential Solutions

    Now, based on the two competing optimization problems I have shown above, it is time to try and find an ideal solution. The first thing to consider, however, is what our base unit is. How small do we want our maps to go? 1 mile hexes are common, but there are several instances where hexes as small as 200 yards across (~1/9th mile) are called for. I will be proposing a few solutions:

220 Yard Base

    First, lets look at a 220 yard base (also known as 1 furlong according to wikipedia). This is similar to the 200 yards mentioned previously, but it is an even 1/8th of a mile. This, coincidentally, evenly tiles in a semi-perfect manner up to a 1 mile hex. From there, considering we are using semi-perfect tiling, nicely tiles 5x into an 5 mile hex. This is the standard hex in which most mapping will be done. For the larger, atlas or campaign hex, tiling 5x to get a 25 mile hex will work well. 

220 yard (x8)> 1 mile (x5)> 5 mile (x5)> 25 mile

176 Yard Base

This one is a bit more of a joke, but it does tile nicely 10x into a 1 mile hex.  This means we are using ideal tiling. From there, we would work our way up to using a 7 mile hex for the main map and then a 28 mile hex for the atlas/campaign hex.

176 yard (x10)> 1 mile (x7)> 7 mile (x4)> 28 mile 

6 Mile Base

    Coming at this from a different perspective, lets see what finagling it will take to have a 6 mile hex as a base, since 6 is a lot more easily divisible than 5 or 7. Going up to an atlas or campaign hex, we could have a 24 mile hex or a 30 mile hex depending on if we prefer ideal or semi-perfect tiling. Going down, however, gets a little more iffy. Because of how the math as worked out, an ideal tiling solution will always be one tile past or 2 tiles short of a perfect 6 and a semi-perfect solution will always be 2 tiles past or 1 tile short. Effectively, this means we have two options: stick with our nice tiling and get a wonky number for the lower distance, or get a nice number but tile in an unpleasant way. 

    Lets start with he first option. Using 4x, we get a smaller size of 1.5 miles per hex and using 5x we get 1.2 miles per hex. Coincidentally, both of these are approximately what might be termed a half-league. A League being an old form of measurement defined as how far someone can walk in an hour. This has wonderful properties for an RPG, since it makes calculating time fairly trivial. If we define the smaller hexes as half-leagues, someone would expect to be able to walk 24 miles per day or 19.2 miles per day depending on which you used. Both are reasonable amounts, though I personally prefer the first since it syncs perfectly with 6-mile hexes. The other makes any further work downward even more ugly, so I will be stopping with the semi-perfect approach here. If we choose to go even smaller, 264 yard hexes perfectly tile 10 times to become a 1.5 mile hex. 

264 yard (x10)> 1.5 mile (0.5 League) (x4)> 6 mile (x4)> 24 mile

    Now, let us consider the alternative of having nice numbers that tile in an unpleasant way. The most obvious manner of doing this would be to go down to a 1 mile hex (6x), which then can sync into your preferred manner of handling the smaller yard units.

220 yard (x8)> 1 mile (x6)> 6 mile (x5)> 30 mile

176 yard (x10)> 1 mile (x6)> 6 mile (x4)> 24 mile

Final Evaluation

   After doing all of those calculations I believe the best solutions are the 220 yard base, or the 6 mile base that takes advantage of leagues. The first has the distinct advantage of being powers of 5 for the most part, with nice numbers all around. The other uses leagues which I believe is an advantage not to be overlooked, since it makes the calculation of travel times trivial in many situations. 

    As for the others, a 176 yard base is just silly and powers of 7 do not mesh particularly well with any other numbers commonly used for travel. It is workable, based on the principles I have laid out, but I do consider it more of a trap option than anything else. The 6-mile base variant does help fix this, but it relies on ugly tiling. The same problems exist for the semi-perfect solution as well.

Personally, I am currently leaning toward the 6-mile base with leagues as the superior option, but my thoughts may change on that matter. I am curious to hear what others thing about it as well.

Addendum

    While going through some blogposts, I came across this blog post about dungeon sizes where the author mentioned having using a 3 mile hex and a 24 mile hex. This, surprisingly, seems to work exceedingly well. Each 3-mile hex could also be called 1-league, meaning that someone could travel 8 hexes per day. 8 tiles is semi-perfect and actually can be made of 8 660 yard subhexes or 20 264 yard subhexes. Obviously it isn't perfect, but it is another interesting option.

264 yard (x20)> 3 mile (1 league) (x8)> 24 mile



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