Grids and Miniature Wargaming ... a never-ending discussion

Introduction

Grids for miniature wargaming are regularly discussed on various forums. Often, such discussions revolve around the procedure for counting distances along the grid. Often, the grids under consideration are either the hexagonal grid, the square grid, or the offset square grid (the so-called brick pattern, which is topologically equivalent to the hexagonal grid for most purposes).

However, there are many other types of grids. In mathematics, grids are well studied and also referred to as "tesselations" or "tilings of the plane". Different constraints can be put on such grids: should all grid cells have the same size and/or shape? Should the grid have a repeating pattern? Should the grid never have a repeating pattern? See this wikipedia page for an introduction to the topic.

Just to get your brain juices flowing, here are some specific pentagonal tilings (a pentagonal tiling uses pentagons). Would it be possible to use such a tiling for a miniature wargame?

The 15th monohedral convex pentagonal type, discovered in 2015

The "Cairo" pentagonal tiling.

Many of you would shudder at using such a grid, but why is that exactly? After all, there are plenty of examples of irregular grids that have been used in board wargames, also known as "area-based  movement" to distinguish them from "hex-based  movement".

Storm over Arnhem often is credited to be one of the first board wargames to use area movement, but a boardgame classic such as Risk uses area movement as well, as do countless other board(war)games.

Grid used in Storm over Arnhem. Image from Boardgamegeek.com

Risk map. Image from Boardgamegeek.com

Why do we not consider such playing grids for miniature wargames, and typically restrict ourselves to either a hexagonal grid or a square grid? After all, it should be easy enough to place miniatures in a grid cell, and move them from cell to cell, just as we do in such boardgames?

For miniature wargaming, we often need more functionality from the grid than simply moving playing pieces from cell to cell. More specifically, we need the following:

• A movement procedure for miniatures or units on the grid;
• A procedure for determining shooting ranges;
• A way to orient miniatures or units relative to the orientation of the grid;
• Align units to adjacent gridcells, such that we can make linear battlelines;

We will discuss each of these issues below.

Movement on the grid

This is the topic that usually gets most of the attention when discussing grids for miniature wargaming. Often, people try to come up with ways to move units on a square grid such that the distortion for diagonal movement is corrected. See also the previous blogposts Square Grids and Square Grids (2) on the topic, in which I also explain that we do not want a measurement procedure (measure a movement distance from starting cell to end cell), but rather want a counting procedure (count expended movement points when moving one cell to an adjacent cell).

In principle, it's very easy to come up with a counting procedure - simply count the number of cells as you move along. However, we want to take into account the various connections between cells. If connections are not symmetric (as in the case of a square grid), movement might become a bit more complex. In irregular-shaped grids, it might become very complex.

Movement on the Cairo grid. Each cell counts as 1 movement point, only edge-to-edge movement allowed.

Movement on the Cairo grid. Each cell counts as 1 movement point, edge-to-edge and point-to-point movement allowed.

Then why do some board(war)games use irregular grids? Usually, because movement is restricted to moving only 1 area, or perhaps 2. In such cases, the total movement distortion when compared to the "true" Euclidean distance is less of an issue.

Moreover, the irregular shaped cells can often serve a purpose. Difficult terrain can be turned into smaller grid cells, and easy-going terrain into larger grid cells, thereby avoiding different movement point costs for different types of terrain.

In miniature wargames, we are so used to having movement speeds doubled or halved depending on terrain, that we usually don't consider irregular grids for that purpose. Often, we prefer regular-shaped grids, and stick to different movement points for different types of terrain. But this also has a reason. Miniature wargames - unlike board wargames - often employ a different terrain setup for each game. Having your irregular grid reflect the terrain sounds like a great idea if you have a fixed map for each and every game, but when you want to shuffle terrain around for each game, a practical solution is not immediately feasible. However, this should not prohibit us from using irregular grids, since different movement values depending on terrain in a specific grid cell is still a possibility.

Shooting ranges on a grid

Most miniature wargaming rules require us to measure the distance between a shooter and a target. Again, as in a movement procedure, we rather want a counting procedure rather than a measurement procedure. We usually want to be able to count the number of cells that lie between the shooter and the target, and use this number as the shooting distance to determine whether the target is in range, whether modifiers need to be applied, and so on.

This is the real bottleneck for using irregular-shaped grids in miniature wargaming. Although we can imagine counting the number of cells, on an irregular grid we might be left to wonder whether it is the shortest distance possible. Especially when the size of the gridcells reflect the type of terrain as mentioned above, the counted shooting ranges can become really distorted, and it would allow you to shoot further if the intermediate terrain is easy-going and suddenly reduce your range when you difficult-to-traverse grid cells lying in front of you. Hence, counting shooting ranges requires cells more or less of equal size.

However, if your ground-scale is such that shooting is restricted to adjacent cells, this is not really a strong requirement. Some distortion might pop up, but no more as in the many boardgames that use an irregular grid and allow adjacent combat only.

Related to determining the shooting range is the issue of visibility. On hexagonal or square grids, the line-of-sight is checked vs intermediate grid cells and terrain therein that might block the line of sight. Because of the regularity of the grid, deciding what cells are crossed by the shooting line can often be eye-balled. But not so in an irregular grid, where this would become more complex, unless you limit shooting ranges to 1 or 2 cells.

Orientation of a unit within a gridcell

Miniature wargames often stipulate firing arcs for units when shooting. When playing on a grid, this means positioning units in a specific orientation on the grid (facing an edge, facing a corner, ...), and defining shooting arcs in terms of grid cells. Often, such a shooting arcs takes the form of a "wedge". In the case of hexagonal and square grids, this is often straightforward, but for irregular grids, this again is a non-trivial procedure if your shooting range extends to 2 cells or more. Even a shooting arc of 180 degrees becomes non-trivial to determine.

Alignment of a unit to adjacent grid cells

Another issue that has to with alignment, is the alignment of adjacent cells, and hence adjacent units. Some periods in which linear warfare is a major element on the battlefield, require that you can line up units next to each other. Easy to do on a square grid (at least in the horizontal and vertical direction, and perhaps the diagonal one), a bit less easy to do an a hexagonal grid (although there are 3 main axes each at 60 degrees where this is possible, but not orthogonal), but almost an impossibility if you use an irregular grid.

However, if the game is a skirmish game (no lineair formations needed), or set in a modern period (spread-out troops), this is less of an issue.

Conclusion

Taking all of the above into account, we want a grid that:

• has uniform, regular, more-or-less equal-sized cells, such that we can have an easy counting procedure.
• allows for easy orientation of units inside a cell and alignment with adjacent cells.

This brings us mathematically to uniform convex tilings, tilings which consist of regular polygons. When we take a look at the list of these tilings , we encounter the usual hexagonal and square grids, but there is also at least one other tiling which might prove to be useful to miniature wargaming, but which has (at least to my knowledge) not really been explored: the triangular tiling.

Triangular tiling

I think the triangular grid has a number of unexplored advantages, not in the least advantages in terms of alignment. However, it has asymmetric connections (both edge-to-edge and point-to-point) which might make a counting procedure more difficult. But we'll keep a full analysis for a future blogpost!

Addendum

1. As can be expected, the discussion of grids (and especially hexagonal grids) has a long tradition in board wargaming. See e.g. this discussion on boardgamegeek. 
2. I also wrote a follow-up post n triangular grids.