Wikipedia:Reference desk/Archives/Mathematics/2017 February 26

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February 26

Infantry squares field of fire

I was watching the film Waterloo last night. Really impressive battle scenes, nothing done with CGI, thousands of soviet soldiers used as extras apparently. I was wondering about the fields of fire from the British infantry squares which were formed when they were under cavalry attack. The squares are portrayed as being very close together and some seem to be directly firing on each other. See this still from the film. Is there a mathematical solution to the best orientation of the squares to avoid "friendly fire"? The solution is easy for a single line of soldiers, a line of diamond orientated squares like: ◇◇◇◇. However, in the film (and probably in reality as well) the squares extended to a depth of three or four lines. This is much more tricky to solve. Arbitrarily reducing the density of squares can produce a solution, but the density of squares has to be taken as fixed by the density of the troops while they were in-line formation. I don't believe any solution is possible for an arbitrary number of lines and infinite range of weapons, so the range of the rifles has to be taken as a parameter. Alternating rows of diamonds and squares works quite well,

◇   ◇   ◇   ◇
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Rows of diamonds works even better for up to three rows if they can be spaced wide enough to allow the squares in the two rows behind to fire between them. That is a reasonable spacing on the assumption that the in-line length of the formation is 4× the side of the square.

◇    ◇    ◇    ◇
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Is there a better solution? I guess this counts as a kind of packing problem. SpinningSpark 13:32, 26 February 2017 (UTC)

I guess the first question would be what do you mean by "better"? Are you're trying to pack the soldiers as densely as possible, do you want to maximize the area covered, or do you want to maximize the chances of winning a battle based on some model of combat? As you mentioned above, the range of the weapons has to be considered, but there are two parts to that. First, what is the effective range to hit your target, and second, how far can a shot be expected to travel and possibly hit an ally if it misses the enemy. Another factor would be how disciplined your soldiers are, since it's one thing to tell them to only shoot straight ahead, but it's another to have them actually do it when there's a armed horseman coming at them at an angle. Not having engaged in early 19th century warfare myself, I's sure there are many other realities of the situation that I simply can't imagine.

But if you reduce the problem to it's simplest possible form you might phrase it in terms of chess pieces. Namely, how many rooks and bishops can be placed on an nxn board without attacking one another? For rooks only the answer is trivially n. For bishops only it seems you can do a bit better - 4 on a 3x3 board, 6 on a 4x4 board and in fact the answer is known to be 2n-2, see [1]. --RDBury (talk) 01:45, 27 February 2017 (UTC)

One way to pack in more firepower without having them shoot each other is to fire from different heights. The front row would lie down, the 2nd row kneel, and the third row stand. Thus, the back rows fired over the heads of those in front. If they are prepared to fire to the rear, as well, that means 6 rows of soldiers, and perhaps more in the middle in support roles, such as replenishing ammunition, caring for the wounded, replacing fallen soldiers, etc. And if you have a natural elevation difference, such as a hill, even more rows can fire over the heads of those in front. StuRat (talk) 04:39, 27 February 2017 (UTC)

• Not a mathematics answer, but my memories of The Mask of Command by John Keegan suggest me a hypothesis to the "friendly fire" question. (One big thesis of that book is that the increasing firearm range between Napoleon and the Civil War lead to a radically different style of command, because generals on the battlefield would be at a much higher risk of being targeted by sniper rifle in the latter than in the former.)

See Napoleonic_weaponry_and_warfare#Firearms: rifles only started to appear during the Napoleonic wars and were apparently reserved for semi-elite marksmen. The common soldier had a musket without rifling; from the various linked articles I gather that the effective range of non-rifled firearms was about 100-200 meters, after which the bullet deviated too much.

If friendly-fire casualties beyond 200m were negligible because almost all bullets would run in the ground or slow down too much by friction, separating the infantry squares by that distance would guarantee they could always shoot without any risk of hitting one another. Tigraan^{Click here to contact me} 12:41, 27 February 2017 (UTC)

We should also look at it strictly from a cost v. reward basis. The naive answer here is that if shooting in one direction is more likely to hit an enemy than your own forces, it's worth it. However, this would only work if both sides had equivalent forces. A more sophisticated formulation would be to say, if your side takes 1/10th the casualties of the enemy, due to superior equipment, tactics, training, numbers, etc., then you only want to fire if you are 10 times as likely to hit the enemy as your own forces. If course, this is only a rule of thumb, and specific situations will change that logic. For example, if an enemy is approaching with a bomb that could wipe out your entire infantry square, you'd better shoot at him, even if your own forces are directly behind him. StuRat (talk) 15:06, 27 February 2017 (UTC)

I'm not really looking for a practical military solution. If I was, I wouldn't have asked the question at this board. Perhaps I have not been clear enough. The problem is to devise a packing that maximises the perpendicular distance from the square edges to any other square while maintaining the overall density of troops equal to the in-line formation. Arbitrarily setting the squares 200 metres apart is therefore going outside the problem boundaries (and I believe it is also militarily impractical). The line density of troops in-line and in square edges should be assumed equal (or at least proportional). According to Line (formation) a line consisted of two to five ranks with three being most common, and, according to Infantry square, Wellington's squares at Waterloo were 500 men in four ranks in squares less than 20 metres on a side. A distance between lines is also needed. I have assumed this to be just enough to allow squares to form and have taken all lines to have the same number of ranks. SpinningSpark 15:17, 27 February 2017 (UTC)