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:
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| Half hexes around perimeter. |
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| Workaround for half 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
Perfect Master Hexes - Category I
The properties unique to Category I:- They have a center sub-hex as can be seen in the images below.
- They will have an odd number of hex rows that pass through a face.
- The six vertices of the Master Hex have the same form within a type.
- The Type A has a whole hex at each vertex.
- The Type B has two half hexes at each vertex.
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| Type A Master Hex |
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| Type B Master Hex |
Imperfect Master Hexes - Category II
Properties of the Category II Master Hexes- The center of the Master Hex is where the vertices of three adjacent hexes meet.
- The number of hex rows passing through a face will be an even number.
- 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)
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 do alternating shapes, you don't have any overlaps.
ReplyDeleteFor 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.