The Joys of Triangles
A Romance of Mostly Two Dimensions.
'Twas thine alone, O youth of giant frame,
Isosceles! that rebel heart to tame!
In vain coy Mathesis thy presence flies:
Still turn her fond hallucinating eyes;
Thrills with Galvanic fires each tortuous nerve,
Throb her blue veins, and dies her cold reserve.
— Yet strives the fair, till in the giant's breast
She sees the mutual passion's flame confessed:
Where'er he moves, she sees his tall limbs trace
Internal Angles equal at the base;
Again she doubts him: but produced at will,
She sees th’ external Angles equal still.1
Navigating the world requires some wrestling with “is” and “ought”. Each a terribly interesting, and unresolvable puzzle. Fortunately, St Barbara has blessed us with an array of handy tools to sail this sea of complexity. One such delightful tool is the humble triangle.
When we flit between principle and pragmatism, the triangle is there as a loyal guide. When we weigh our judgement between one factor and another, she breaks us out of our one-dimensional romance, and tells us a thing or two about the intricacies of our world.
The triangle serves to apply some geometric honesty to our balancing of factors and delivers a constant reminder of the maxim of maxims: everything has a price.2
At risk of reaching a Limit of Exploitation on metaphors: if tactical thought is a river, then limiting ourselves to the seesaw of ‘balancing’ two extremes will only yield an In-Place force. By weighing three factors and comparing them with a spatial value, triangles offer the establishment of a Bridgehead of understanding. With a little time and the proper investment, we might well secure the far bank, and give the opportunity for a Break Out.3
As with all crossings, while we must give exploitation on the other side much consideration, since it is the point of crossing in the first place, we cannot know what triumph or tragedy awaits us until we have at least got our feet wet.
All great Military Theorists love triangles. This is known.
Jomini is proof that you can do all the diagrams you want, have a reputation as the original (military) geometry gangsta, even popularise important insights, but you might as well not bother these days if you don’t get amongst some triangles. Clausewitz, to the best of my knowledge, was without triangles, but he still respected three sided-thought and, above all, was the trinitarian (and that is all the explanation we need).
One of the triangular greats, a true triangle connoisseur, is Brigadier Richard Simpkin MC. A harbinger of several British Cold War armoured vehicles and a dab hand at theoretical meanderings, he was a man that loved triangles.
Simpkin makes a number of deep forays into equipment, with this triangular illustration imparting the emphasis different nations have placed, intentionally or otherwise, on the designs of their tanks.4 Here, we can see that the UK is most persuaded by the virtues of firepower and protection, while the USSR comes nearer the center, with a bias against protection in favour of mobility and firepower.
Simpkin then investigates mobility. And it is here that Simpkin teaches us something about ‘nesting’. That is, the mobility element within the marketing man’s triangle simplifies another interesting relationship, that of the tradeoffs within mobility itself:
operational mobility, the ability to cover long distances fast and reliably on roads;
tactical or off-road mobility, concerned with movement out of contact within the combat zone using minor roads, trails and selected crosscountry routes;
battlefield or cross-country mobility, meaning speed and freedom of maneuver in contact.5
‘Operational Mobility’ and ‘Battlefield Mobility’ are each in tension with ‘Protection’. The UK design of the time (Chieftan) favoured Protection over Mobility (slightly biasing Battlefield Mobility over Operational Mobility though).6
We can clearly imagine this tension – you can drive far and fast, but it costs you the ability to manoeuvre tactically over varied terrain in an ‘agile’ manner. These, in turn, are not achievable if you bump up your weight (read weight as Protection).
Simpkin then illustrates further nested tradeoffs with a triangle of a more technical nature:7
Within Battlefield Mobility, we have the properties of ‘Agility’ and ‘Mobiquity’ (that is best described as the proportion of traversable terrain). As Operational Mobility is akin to ‘Road Speed’ (stand fast transportability and obstacles crossings), we have this new triangle showing the tensions of ‘Agility’, ‘Road Speed’ and ‘Mobiquity’.
Agility, mostly a function of break horse power per ton, does provide a helping hand to Mobiquity, mostly a function of ground pressure (and some track ratios), but each strain against Operational Mobility, which is mostly a function of road speed (and reliability).8
Another triangular warrior is Professor Jim Storr, a prolific writer in the British Army Review of the 90s and noughties. Here he illustrates some properties of a WW2 Panther as a reference point. It is deliberately depicted as a ‘balance’ of mobility (horsepower per ton) and protective and penetrative qualities against rolled homogenous armour equivalency (RHAe):9
So far so good, but then Storr takes these axes and bursts the geometric dam:10
This is Storr, in his effervescent chaotic-good manner, bursting out of conventional constraints.11 General Sir Henry Shrapnel would be proud.
What Storr imparts on us is an insight into by whose standard? Of course we can plot things inside a triangle and illustrate a few trade offs, but how does this meaningfully compare to yesteryear? This isn’t just insight for historians either, it is also for the rampant empirical pragmatists of today. One need not be staring down the barrel of a former museum piece to know that the gap between old and new might merely be a few sighting systems.
Then there is Lieutenant Colonel Robert Leonhard. A man who can seduce you into loving manoeuvre theory, before he teases you with time, which was really just a setup to entrap you into an affair with the information age. While many theorists might be accused of being a man-of-one-book, Mr Leonhard is that, but three times over. Thus he lulls you into a false sense of security as each of his works is masterfully coherent and comprehensive on their own, but dares you to reconcile the connections between them.
Although he more frequently makes use of cartoonish analogous diagrams to illustrate his excellent points, he has also flirted with triangles - taking on a more abstract style than others:12
Here, Leonhard is enlightening us on a relationship between possible states. A protective state is suboptimal for moving, given it is, abstractly speaking, striving to be orientated everywhere rather than going somewhere. Each relationship is opposed to the others. While it might be possible to deploy in a half way position between two states, that organisation could not achieve the properties of the opposing third state explicitly.
And this relationship between is a key triangular insight from Leonhard. He teaches us humility in our triangle dealings, saying don’t forget about the sides:13
Besides arming us with a frame to look at wider theory, Leonhard has also armed us with something to say about the original Mobility-Firepower-Protection triangle. A nation which leans towards a Mobility-Firepower pairing, is declaring that Protection is an implicit property that will be created by lots of moving quickly and lots of firing loudly. Likewise, an emphasis on a Firepower-Protection alliance, seeks to achieve Mobility from all of the freedom derived from lumbering around with a stout sword and shield. Needless to say, a combination of Mobility and Protection is its own form of Firepower.
One instructor of the artillery sort produced this deceptively handy triangle to communicate the woes of Gunners to the other arms:
Here, in an effort to enhance cooperation, this illustration conveys some of the frictions of planning artillery fires. A quick fireplan costs you accuracy, while an accurate one costs you time. Of course there are different circumstances that merit the focus of each. Efficiency, something not always desirable in Gunnery, is usually a feature of lengthy operations such as in, where we first began, obstacle crossings.
So with the military might of flatland, we might be able to make further theoretical progress. It is no coincidence that in Edwin Abbot’s portrayal of Flatland, he depicts his foot soldiers as possessing definitive points:
“…but most ravishing of all is said to have been the unspeakable magnificence of a military review.
The sight of a line of battle of twenty thousand Isosceles suddenly facing about, and exchanging the sombre black of their bases for the orange and purple of the two sides including their acute angle; the militia of the Equilateral Triangles tricoloured in red, white, and blue; the mauve, ultra-marine, gamboge, and burnt umber of the Square artillerymen rapidly rotating near their vermilion guns; the dashing and flashing of the five-coloured and six-coloured Pentagons and Hexagons careering across the field in their offices of surgeons, geometricians and aides-de-camp - all these may well have been sufficient to render credible the famous story how an illustrious Circle, overcome by the artistic beauty of the forces under his command, threw aside his marshal’s baton and his royal crown, exclaiming that he henceforth exchanged them for the artist’s pencil.”14
Our world, but especially the Profession of Arms, is permeated by paradox. The Joys of Triangles arms our soldiers for these paradoxes and tradeoffs. It tempts them to (temporarily) throw aside their arms, and pick up the theorists’ straight edge. Coupled with a protractor and compass, we might well establish that Bridgehead.
The Loves of the Triangles: A Mathematical and Philosophical Poem (1798) by C Frere.
For more on things-are-what-they-are and are not-what-they-are-not, consider weakness.
Astute doctrine monks will note that this is a British crossing, as an American one would include obscuration as a matter of principle - which would not be conducive to understanding.
Tank Warfare (1979) by Richard Simpkin, page 82.
Ibid, page 107.
I’ve heard a few people remark on how Chieftains switching through gears made for some interesting constraints on Forming Up Point selection and use.
Ibid, page 100.
And water obstacles!
Battlegroup! The Lessons of the Unfought Battles of the Cold War (2021) by Jim Storr, page 96.
Ibid, page 103.
He also has a superb digression on enfilade, using the medium of our favourite shape.
Fighting By Minutes: Time and the Art of War (2017) by Robert R Leonhard, page 24.
Ibid, page 30.
Flatland: A Romance of Many Dimensions (1884) by Edwin A Abbott, pages 26-27.
Can you give us some geometry 📐 for the Drones? Seriously.
Wunderbar.