Wednesday, March 8, 2017

Mapping with Master Hexes

Hex grids have been used for mapping in gaming since the 2nd edition of Gettysburg by Avalon Hill in 1961.  DragonQuest, published by a company known for its war games, uses the hex grid for tactical and campaign mapping though it wasn't the first RPG to do so.  This post is going to focus on the use of what I'm calling Master Hexes for mapping and isn't specific to DragonQuest.  This concept has been around since the very early days of the Judges Guild (Campaign Hexagon Sub-System) and maybe longer. I'm making no claim on having come up with anything particularly original here.  The inspiration for this post came from several blog posts that I came across recently.
The basic concept is simple - you take a hex that represents some large area and sub-divide it up into many smaller hexes.  So, for example, if you had a campaign map hex that was 5 miles across and you wanted to get down into the nitty gritty detail you might choose to split that 5 miles across into 25 sub-hexes that would be 0.2 miles across.  One issue that arises when you do this is that your Master Hex ends up with a number of the sub-hexes along the sides that are half in and half out.  Like this:
Half hexes around perimeter.
Unfortunately there isn't anything to be done about that and retain a perfect hexagonal shape for your Master Hex.  In general this isn't much of an issue though you do have sub-hexes which belong to 2 different Master Hexes.  A clunky workaround that I don't recommend is to include the full hexes on alternating faces of the Master Hex like this.
Workaround for half hexes.
This awkward shape then nests perfectly with adjacent Master Hexes.
If you want sub-divide a hex such that the width between faces is an integer number of sub-hexes you'll see patterns emerge.  I have split these into three types of Master Hexes which belong to one of two categories.
  • Category I - Perfect Master Hexes
    • Type A
    • Type B
  • Category II - Imperfect Master Hexes
    • Type I
Each category and type have different properties and I'll go over each one.

Perfect Master Hexes - Category I

The properties unique to Category I:
  1. They have a center sub-hex as can be seen in the images below.
  2. They will have an odd number of hex rows that pass through a face.
  3. The six vertices of the Master Hex have the same form within a type.
    1. The Type A has a whole hex at each vertex.
    2. The Type B has two half hexes at each vertex.
Type A Master Hex
Type B Master Hex

Imperfect Master Hexes - Category II

Properties of the Category II Master Hexes
  1. The center of the Master Hex is where the vertices of three adjacent hexes meet.
  2. The number of hex rows passing through a face will be an even number.
  3. The vertices of the Master Hex will alternate between a whole hex and two half hexes.

Sizes of Master Hexes

Each Type gives a specific set of sizes and from the images you can probably see that the sizes for each type follow a pattern.  These are the formulas for the size progression of each type where i is an integer and is the ith size of that type.
Type A: 3i + 1 giving sizes of 4, 7, 10, 13, ...
Type B: 3i + 2 giving sizes of 5, 8, 11, 14, ...
Type I: 3i giving sizes of 3, 6, 9, 12, ...

Why Perfect & Imperfect

Property 1 of Category I means that as you zoom in to the center of a hex the sides of that hex become the sides of the encompassing Master Hex.  This is not true for Category II Master Hexes.
Because of Property 1 of Category I, if you move in one of the six cardinal hex directions from the center sub-hex you will be moving in a straight line through the centers of adjacent hexes and will pass through the center of the Master Hex side and on into the center of the next one (see the light green highlighted cells).
I find the 6 whole sub-hexes at the vertices of the Type A Master Hexes more aesthetically pleasing.  Also those six hexes are distinctly within the Master Hex.
I can think of only one benefit to the Category II, or Imperfect Master Hexes, and for those that ascribe to the 6-mile Hex school it may be significant.  Category II hexes come in widths of 6, 12 and 24 giving you sub-hex sizes of 1.0, 0.5 and 0.25 miles.
Type A Master Hexes give you widths of 10 and 25 which works well for 5-mile Master Hexes giving 0.5 and 0.2 mile sub-hexes. The size 10 Master also works well for metric systems or when you just want to use factors of 10.  I believe the Judges Guild blank mapping pages used the size 25 Type A Master Hexes.
Type B Master Hexes give you widths of 5 and 20 but also 50 sub-hexes which, for 5-mile Masters, gives you 1.0, 0.25 and 0.1 mile sub-hexes.

Master Hexes by the Numbers

The area of a Master Hex, expressed as sub-hexes, is the square of the width in sub-hexes. So a 10 hex wide Master has an area of 100 sub-hexes.
The number of whole sub-hexes (those that aren't on borders) is different for each type. Where 'n' is the width in sub-hexes:
  • Type A: (n-1)^2 + n
  • Type B: (n-1)^2 + n - 2
  • Type I: n^2 - n or n(n-1)
I don't know that either of those are useful values to have but if you wanted to know without counting the sub-hexes you have the formulas.
Another interesting if questionably useful factoid is that the length of the side of a Master Hex can be expressed in multiples of the length of the side of a sub-hex.  This turns out to be the width of the Master in sub-hexes.  It was from this post (A Foxs Guide to Geomorphs, part 12) that I learned of this and it started me down this whole path.
Last little tidbit.  The number of rows that pass through a Master Hex side can also be calculated from the width(n) in sub-hexes.
  • Type A:  (2n + 1)/3
  • Type B:  (2n - 1)/3
  • Type I:  2 * (n/3)
If you know of other takes on this Master Hex concept, please leave a link in the comments below.

Links to other Hex Posts

Here are some other blog posts about hexes in general and for mapping.

1 comment:

  1. If you do alternating shapes, you don't have any overlaps.

    For example, a World Map with Flat-Top hexagons that when zoomed in, each tile can be an odd number of Pointy-Top hexagons. There are no overlapping.