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Tuesday, 7 August 2018

Triangular Grids

I am a fan of using grids for miniature wargaming. My wargaming table has a permanent Hexon terrain system laid out, but I've also played games on a rectangular grid, most notably my Flagstone Fleets game. See also the Table of Contents for previous discussions about grids.

In this blogpost, I would like to discuss the triangular grid. I think it's a grid that has not been fully explored in wargaming (hexagonal grids and square grids are getting much more attention), but it is worthwhile to look at some of its advantages and disadvantages.

Relation between the triangular and hexagonal grid

There is a strong relation between the classic hexagonal tiling, and a triangular tiling.


Mathematically, they are each other's dual tiling. This means that if you take the centre points of each hexagon, and you connect these centre points together, you get the triangular tiling. This also works in the other direction: if you take the centre points of each triangle, and connecting them together, you get the hexagonal tiling.

This duality results in a nice property: when playing a game on a triangular grid: placing the pieces on the corner points and moving them along edges, is equivalent to playing that game on a hexagonal grid by placing the pieces inside the hexagons and moving them across edges. And vice versa, playing a game on the corners and along edges of a hexagonal grid, is equivalent to putting the pieces in the triangles and moving them across edge boundaries.

There is however a major issue with the triangular grid, which is the asymmetry of connections between the triangles. Unlike a hexagon grid, triangle can connect edge-to-edge, but also point-to-point in various configurations. THis is also the case in a square grid, but in a triangular grid there are different ways in which triangles can connect point-to-point. This lack of symmetry is the most likely reason why triangular tiles have not been really considered for wargaming.

Has the triangular grid been used before?

When trying to look up whether (board) wargames make use of a triangular grid, surprisingly few results turn up. Most of the games that use a triangular grid, use the dual property, effectively using a hexagonal grid. The triangular grid is then merely an esthetic element in the design of the game. See e.g. games that employ the "Triangle System", See also this page at Forsage Games.

Most games that use the triangles themselves as areas usually are abstract boardgames. Some examples are: Blokus Trigon or Go played on a triangular grid.

Counting distances

Let us assume we do want to design a miniature wargame using a triangular grid. One of the things we need is a counting procedure for counting distances from one grid cell to another cell, and preferably, we would like that counting procedure to approximate the Euclidean distance between the centre points of both grid cells.

Using some simple goniometry, and setting the distance between the (barycentric) centre points between 2 adjoining triangles to 1, we arrive at the following relations:


Taking the red dot as starting point, we see that we can get to edge-to-edge triangles (yellow dots) using distance 1. The triangle directly opposite (blue dot) requires distance 2. Both triangles that touch the starting triangle, but are not directly opposite (yellow dots), are at a distance equal to the square root of 3, or 1.73.

Rounding up 1.73 to 2, we get the following, rather simple, counting procedure for measuring distances:
  • When moving from edge-to-edge, count 1;
  • When moving from point-to-point (any configuration), count 2.
This looks like a nice and relatively simple procedure for counting distances. Another way to look at this, is to construct a hexagonal grid on top of the triangular grid, by inscribing each triangle with a hexagon, and putting additional hexagons at the corner points (note that this is a different hexagonal grid as opposed to the dual grid as described above):

(Image from "20 Fun Grid Facts (Hexgrids)")
The yellow and blue triangles (or the yellow and blue hexagons) form our triangular grid. When moving from a yellow to an adjacent blue hexagon (i.e. moving on the triangular grid from a yellow to a blue triangle), you can do that directly, at a cost of 1 movement point. But when you move to a hexagon whose corresponding triangle is touching at a corner, you have to pass through a green corner hexagon, resulting in 2 movement points if you would count along the hexagonal grid. So, our distance rules derived above can also be considered as moving on the underlying hexagonal grid - albeit by disregarding the green hexagons as shown in the diagram above.
(You might wonder how this is possible, given that we have rounded one distance 1.73 to 2 ... but there are also 2 modes of moving on a hexagonal grid: straight ahead from hexagon to hexagon, or in a "zig-zag" pattern, which correspond to our differently connected triangles ... these two movement paths are not exactly equal, although we often consider them as such on the hexagonal grid.)

Ok, let's put it all together. How far can we move on a triangular grid given various amounts of movement points? The diagram below illustrate the movement ranges, using 1 movement point to move across an edge, 2 movement points to move across a corner.

The blue, green and orange triangles indicate the range using 3, 5 and 7 movement points. The dotted black arcs have radius equal to 3, 5 and 7. The dotted red arcs are scaled with a cosine(30 degrees) factor. This allows comparing the "zig-zag" movement when one would move along the horizontal row of adjacent triangles.

Facing, Battlelines and firing arcs

When located in a triangle, a unit can be oriented in 12 different directions: 3 sides, 3 corners, but there also 6 other directions that line up with rows of triangles (see diagram below). This is not unlike a square grid where you have 8 natural facings, or a hexagonal grid that has 12 facings (6 edges + 6 corners).
Square grids have 3 natural main directions along one which can put troops next to each other: vertical, horizontal, and diagonal. A hexagonal grid has 3 main directions, and 3 "zig-zag" directions. What about the triangular grid? The triangular grid also has 6 main directions, as shown below (the 2 other symmetric directions at 30 and 60 degrees are not shown).


The orientation and facings, having 12 "natural" directions on the grid, might be the biggest advantage of the triangular grid. However, just an in squares, the connections between the grid cells is asymmetric. But this is also the case when considering the "zig-zag" directions in a hexagonal grid.

Firing arcs become a little more complex. The diagram below shows firing arcs at 60 degrees, and a distance of 4, using the counting metric as derived above. Note the little discrepancy in the firing arc for the unit on the right.


All this seems workable, but it takes time getting used to.

Conclusion

So, should we use the triangular grid for our miniature wargames? Honestly, I don't know yet. I will have to run a test game or two ... but any other experiences or insights are certainly welcome!

5 comments:

  1. Today I was calculating the size of triangles to fit my bases, and I found a huge (in my opinion) drawback. My bases are 4 х 4 centimetres. Square area is 16 cm. sq. To fit such a base one would need a triangle with square area of 34 cm. sq. So half of the table is some kind of "lost". My gaming area is 100 x 100 cm. I can fit 20 x 20 = 400, 5 cm. squares in it. If I use triangles I would only get 252 zones to play with.
    On the other hand all this empty space inside a triangle leaves enough room for markers. So if one uses a lot of markers to shows different unit statuses it can be quite handy.

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    1. That's a good and valid point I really didn't think about before ... how much useful area you really have in certain types of grid. I agree that when our bases are square, a square grid is a better solution, but for (skirmish) games with single-based figures, it might not matter that much.

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    2. Even for skirmish games triangle is the least efficient figure. For example, I skirmish with figures based on round bases with radius 1.15 mm. In order to fit such a base triangle would have an area of 6.93 cm. sq, hexagon 4.58 cm. sq, and square 4.15 cm. sq.

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    3. That's if you consider 1 figure in 1 gridcell. But in my hexgrid games - using a Kallistra grid, hexes are 10cm across - we allow multiple figures in one hex. Also, if you play let's say ww2 games with single-based figures, you still have squads that move together as a group. Such figures can easily be positioned as a group in a triangle cell. But eprhaps I should do a few experiments with triangles of various sizes.

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    4. I agree. Sometime after I wrote my comment a thought came to me that if rules allow multiple single-based figures per zone than triangle may be more efficient, depending on how many figures are usually allowed per gridcell. The only visual drawback might be that this figures would always be positioned in kind of wedge formation which probably may look a bit strange for some periods.

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