Speaking as a former infantry weapons platoon commander, this looks like a pretty good fire plan.  You employ the guns in pairs (squads) with overlapping fields of fire. MGs may also be assigned a Principle Direction of Fire (PDF) within the field + a Final Protective Fire (FPF) that runs roughly parallel to ones own front line and is used to prevent the enemy from getting into your position.  MGs have a T&E mechanism (Traverse and Elevation) which can be used to set the field of fire on the gun.  You'd probably have more infantry to deal with enemy dismounted infantry + mortars to cover any MG dead space. The LAW gunners by the bridge are pretty exposed. Maybe they take a shot and pull back to the store to help cover the kill circle.    

A lot of connections between military and math. Of course artillerymen use it a lot.  The battlespace has its own geometry of unit boundaries, restrictive and permissive fire control measures, etc.  Then there is the whole field of Operational Research.  Military officers uniquely have to think in a big, complex, 4-dimensional box crowded with forces, lines and objects of various kinds. A good book is MIT Press The Heirs of Archimedes: Science and the Art of War Through the Enlightenment.     Wish I'd had you for a math teacher!     

I have done quite a bit of writing, adding 15 pages to my manuscript.  The machine gun emplacement section is now on pages 79 to 81 and I have added an additional scenario:

“Suppose that we have four machine guns and wish to set an ambush from defilade, leading a BMP-2 that will drive across a 50-meter opening where vision is obscured on either side.  Knowing that the 12.7 mm is marginal against vehicles heavier than the BTR-80, the armorer has mounted them in pairs slightly inwards from coaxial so the bullets converge at a range of 100 meters.”

The original scenario remains the same, but that does not mean I am ignoring your comments.  I am just coming back to Research Gate now and looking at them.  I may still modify this scenario to make it more realistic, but let me now defend it as originally written.

Reed Bonadonna-

“The LAW gunners by the bridge are pretty exposed.”

The LAW gunners by the bridge are below the -4° minimum elevation angle of the BTR-80 auto-cannon, but they are exposed to small arms fire from dismounted infantry on the bridge.  Except for MG3, the M2 gunners are outside the range of small arms such as the LPO-97 (the greatest fear of the LAW gunners by the bridge) and are most exposed to the 14.5mm auto-cannon of the BTR-80.

Combined arms doctrine requires presenting the enemy with a dilemma by engaging them with two different weapons at two different ranges.  The 14.5mm auto-cannon has crude optical sights and so the enemy would attack in daylight and immediately storm the bridge with dismounted infantry led by a BTR-80 firing forwards at MG3.  They are not confident that they can hit the other M2 gunners, but they can drive fast so it is hard for the M2 gunners to hit them.  Thus, MG3 is most exposed, but somebody has got to be the hero who stands directly in front of the enemy – you can’t all be hiding around the periphery.

If the roles were reversed and it were a Western army attacking with the LAV-150, they would come at night and linger in the hills while methodically killing all the Kord gunners from hundreds of meters away using FLIR sights.  The RPG gunners near the bridge would just keep their heads down until they have targets above them on the bridge; they would feel less exposed than the Kord gunners.  The LAV-150 has a -8° minimum elevation angle and only three soldiers who can dismount, so RPG gunners would want to get closer to the bridge than LAW gunners would want.

When using combined arms, it is always true that one type of gunner has a relative advantage compared to the other type, but the real advantage comes from using them simultaneously.  It is by being presented with a dilemma that the enemy is shattered, not by the small advantage that one weapon has over another.

Ross Stewart-Smith-

“look up ‘beaten zone’ with reference to direct fire weapons”

The beaten zone was addressed in footnote #21:

“The kill circle is not what beaten zone meant during WWI.  This term is from the distant past when heavy machine guns were used like low-caliber artillery and fired at a high angle of elevation so the bullets came down at a steep angle of descent.  We are doing direct fire on armored vehicles at close range.”

“the front armor of most modern APCs will defeat a .50 caliber round”

The BTR-80, which this scenario is about, is ubiquitous but it is no longer considered modern.  I spoke to a former Soviet officer who is now fighting for Poroshenko and he said that the M2, with armor-piercing (not ball) ammunition, can defeat the frontal armor of the BTR-80.   But he is an old man whose experience was in the Soviet invasion of Afghanistan when the BTR-80 really was modern.  It is possible that they have better armor now.  He mostly warned against trying to get above it, which was possible against the BTR-70 because it could only elevate its gun to 30°.  Laying ambushes from a height was a tactic that initially worked but then stopped working when the Russians introduced the BTR-80 with its 60° maximum elevation angle.

“you want to avoid having friendlies ‘long’ of any of your weapons”

This is an exercise in geometry and the basic assumption is that the top traverse of the machine guns is a hard limit.  If the top traverse were just a suggestion, then the geometric analysis is pointless because, once the shooting starts, everybody would just be spraying.  If the top traverse is a hard limit, then nobody is long of friendly machine guns; the LAW gunners near the bridge seem to be but they are by the river bank and so the bridge itself protects them from grazing fire on the surface of the bridge, which is well above them.

Surrounding the enemy on level ground is very difficult without being long of your own guns, which is why canyons were so important during the Apache war.  The U.S. Army response back then was to arm their soldiers with accurate single-shot rifles that could hit Apaches high on canyon walls, while the Apaches were only marginally accurate from up there using lever-action carbines.  But the Army didn’t realize that these attacks from elevated positions on both flanks were mostly intended to wound or stampede horses and thus delay pursuit.  When Chief Victorio went to kill the U.S. soldiers (e.g. 24 May 1880 at the entrance to Cooke’s Canyon) he lay his ambush in an arroyo parallel to and only a few meters from the road.  We can assume that the soldiers all had their eyes on the walls of the canyon they were entering and were not looking down, though we cannot say for sure because there were no survivors.

Against modern armor, it is today possible to get above the T-90 with its low 14° maximum elevation angle, but the BTR-80 and, even more so, the ZSU-23-4 are devastating to snipers firing from the balconies of high-rise apartments.  I spoke to a Russian infantryman who fought in Eastern Ukraine.  He boasted that “even professional snipers” were killed by the Shilka (NATO designation:  Zues) and mocked them for constructing sandbagged positions on the 20th to 30th floor of apartment buildings, apparently unaware that no bunker they can build up there can stop a 23mm round.  Thus, against the Russians, I recommend fighting Apache-style from basement windows or from the lowest ground available near elevated freeways or bridges.  Also, I recommend that, if the defenders have only one ATGW, it be aimed at the self-propelled anti-aircraft gun, as should all available mortars (the water jackets on the Shilka’s guns can be damaged by mortar shrapnel) while the T-90 should just be ignored.  The T-90 is a very expensive vehicle and represents ten BTRs that are not on the battlefield – let the Russians drive it around to no effect while you focus on their anti-aircraft guns, even if you have no aircraft of your own.

“I’d recommend you look at planning and deconfliction for close air support (CAS).  There’s more geometry than you can shake a stick at there.”

I will.  I want this book to be useful to soldiers in both America and the Ukraine.  CAS is not something I know very much about because the Russians do not have anything like the Predator drone with its Hellfire missiles, nor very many GBUs; also, they have the nasty habit of executing anybody involved in a friendly fire incident, which means that their pilots never drop bombs or strafe anywhere near their own troops.  It was this reluctance to engage that made the hugging tactic used by the Chechens so effective.  But America relies heavily on CAS and I will let you know if I include a section on this topic in my book. 

At this time, without further study, my view is that CAS is devastating against hillbillies in Afghanistan but of little use when fighting a mechanized army in a city of high-rise apartments (e.g. seizing Bandar Abbas in the event that the Iranians close the Strait of Hormuz) because the A-10 does not dive at a steep enough angle to hit vehicles hugging buildings.  The AC-130 is more useful, but it is an SA-7 magnet – it would not survive for long over Bandar Abbas.  On 31 Jan 1991 an AC-130 was shot down by a single SA-7 missile, an event credited to being out after dusk when the plane became visible in the morning sun.  But today our mechanized enemies – not the hillbillies, of course – have the ability to launch guided missiles in the dark.  Note that Hezbollah never lost the ability to destroy Israeli tanks at night in the 2006 Lebanon War, even after 34 days of getting pounded by huge quantities of munitions. 

I will study the subject of CAS from a geometric point of view and get back to you.

So I am speaking from the most elementary of perspectives on this but I would venture a couple thoughts on this. 

a 3 dimensional model would allow you to better represent this problem in a manner that would be helpful, even for design purposes. 

You should look at machine gun theory, this type of weaponry performs in this manner even at short ranges. 

If you observe the battle maneuvers in the doctrinal handbooks you can find ambush planning and tactics. These are very helpful for seeing how to do things but don't generally say why we do them. A maneuver is generally, from a planning perspective, based on a flanking movement. This is so that we can keep our fields of fire separate from our own people. In your scenario, the Law gunners and MG 3 are uncomfortably close to where the bullets from your other weapon systems are going to be going. That level of risk is going to be too high for me, as I'm going to be the guy emplacing and using that weapon. 

I hope this is helpful as feedback. 

Think of most military movements as an L shape. It never works out in practice but thats what we shoot for (heh) even though it never happens in real life. 

I wanted to expand on the machine gun theory comment. i am not an infantryman by trade, however in my service to become a better machine gunner I was taught the basics of it. A machine gun fires true on the first shot. Think of this as your on target, right where the sights are set. After this first shot, because of barrel physics and recoil, the bullets begin to come out in a spiral. So if the target area was an analog clock the first shot would be the center, with the second shot at the 2, third shot at the 4, fourth shot at the 7 and so on. this is a very crude conceptualization of this first theory but I think you can get where I am coming from. This pattern can be observed even at short distances of 400 yards with the lighter weapons such as the M249. 

The second thing to note is that all bullets travel in an arc. With a machine gun this arc combined with the pattern the bullets come out in create the beaten zone where the enemy troops are (hopefully). I will have to bow out on the exact terminology but I also believe this is the difference between point targets and suppression targets. The Infantry chap up above likely has a bit more experience with this side of the business. 

Thank you for doing this btw. The Academic side of our profession is often neglected and understanding what we are doing and how it works is absolutely awesome. I am truly looking forward to your book and research. 

I now have three independent scenarios.  Your comments are all still on the first scenario.  Without dwelling too much on that, I want to point out that I did not mention this because it seemed obvious, but MG3 will get a little concrete wall to hide behind.  Even if we are mixing the cement in a wheelbarrow and it is still green when the enemy comes, it should stop the errant .50 caliber bullet.  It won't withstand sustained fire from the enemy's 14.5 mm autocannon, but they are not capable of that while moving.  A BTR-80 is not like a tank with stabilization software to keep the gun steady when the vehicle is bouncing around.  It has a top speed of 90k/h or 25 m/s, which means it can cross the bridge in four seconds.  If everybody is hiding around the periphery in what Ross Stewart-Smith calls an ambush and I call harassment, the BTR-80 could speed across the bridge unscathed, drive through the plate glass window of the convenience store and crash through the block wall on the back of the store, at which point their infantry dismounts and kills everybody by shooting them in the back.  That's bad.  But an M2 enfilading it at close range can stop it.  Also, at least some of the available concrete would be well-spent on speed bumps for that bridge - the BTR-80's suspension was not designed by Lotus.  Leaf springs on big rubber tires can get bouncy.

Scenario #2  

We have four machine guns and wish to set an ambush from defilade, leading a BMP-2 that will drive across a 50-meter opening where vision is obscured on either side.  Knowing that the 12.7 mm is marginal against vehicles heavier than the BTR-80, the armorer has mounted them in pairs slightly inwards from coaxial so the bullets converge at a range of 100 meters.

Scenario #3

Here we are fighting either in the past or in a modern country too poor to have any armored vehicles nor many machine guns.  The border is defined by a river.  At the confluence of it and another river that comes from your enemy’s land, you, with only one machine gun, are tasked with defending against enemy marines who might come in small boats and attempt to cross the river under fire.  On your bank of the river you have much concertina wire.  If you can kill half the enemy with your gun, your infantry can fight off the other half at the wire.  However, during peacetime you trade with these people, and so there must be a 50-meter wide break in the wire for a dock.  To protect your gun from mortars, you will build a bunker with a 40° slit.

I forgot to mention, I have added a lot of material to my manuscript, so the machine gun emplacement section is now on pages 93 to 97.  Page numbers are changing all the time so, if you have trouble finding it, it is the first of three appendices to the green belt chapter; the table of contents will tell you where that is.

Also, I have a question:  Is the Kord machine gun ever mounted in coaxial pairs?

The M2 feeds in one side and ejects out the other side, and it is reversible, so it is easy to mount two of them coaxially.  But the Kord feeds in the right and ejects out the right, which I am not aware of being reversible.  This would make a coaxial mount awkward.

Scenario #4: 

A straight fence defines the border between two countries and, on the enemy side, geographic features define a bottleneck.  Given a top traverse of 30°, the kill chord drawn across the bottleneck defines an arc that your gun must be positioned on.  There is a point on the fence, , that one infantry platoon is tasked with defending to the left of and another platoon to the right of.  To avoid the appearance of favoring one platoon over the other, you wish to position your gun so its field of fire covers equal segments of fence to the left and to the right of this point.

The machine gun emplacement section remains the same, but is now on pp. 100-104 as I have added quite a bit to the earlier chapters.

Also, of interest to military men, I solved this problem on p. 135:

A nuclear power plant is to be built in an area where there are three parallel highways. There is concern that terrorists might attack, either by land from one of the highways, or by air. Military posts will be built on each of the highways with infantry that can respond to a ground attack. Also, these posts are to be at the vertices of an equilateral triangle with the power plant at the center, so their anti-aircraft guns provide uniform coverage of the air space above it.

This problem (p. 137) is also of interest to military men:

Your roughly circular island has three fairly evenly spaced port cities connected by straight highways. There are many guns aimed seaward at the cities, but there is concern that the enemy might land troops on the coast and then bring guns to bear on the cities from the hills. The army can quickly move troops on paved roads, but not through the muddy interior. Because there are always traffic jams near the cities, the army has decided to build a fort on each of the three highways and connect them directly to each other with roads through the interior that are restricted to military vehicles. Where should they build?

The machine gun emplacement section is now on pages 161-165 (my page numbers, not RG numbers).

I have responded to some of the criticism above.  In scenario #1, I recognize that there is some danger of machine gun #1 firing under the bridge and hitting the grenadiers on the other side of the bridge.  So I moved it so this cannot happen.  Also, I recognize that machine gun #3 is the most vulnerable because it is directly in front of the approaching vehicles, so I drew in a little wall in front of it and attached to the corner of the building.

This will soon be published so, if you have any comments regarding machine gun emplacement as an application of geometry, please let me know.