Ah, so it can be done, just almost never on purpose 
Was this the scene where the Riders of Rohan appear to break the siege of Minas Tirith? In this scene, the orcs are clearly shaken in the face of the onrushing forces of men and break their lines to a certain extent. I think this may have affected things somewhat.
Why squares and not triangles?
Well, yes, but besides, they are the bad guys. 
I’m thinking that the corners would be to vulnerable. That, and triangles aren’t stackable. You could put two infantry squares next to each other for mutual defense. Triangles, not so much.
Could conceivably work, but the corners would be very crowded and harder to defend. Plus it’s pretty hard to form a triangle quickly - right angles are easier.
And although it wasn’t a cavalry attack, one can’t discuss infantry squares without quoting Kipling - more precisely, Fuzzy-Wuzzy:
*So ‘ere’s to you, Fuzzy-Wuzzy, an’ your friends which are no more,
If we ‘adn’t lost some messmates we would ‘elp you to deplore;
But give an’ take’s the gospel, an’ we’ll call the bargain fair,
For if you 'ave lost more than us, you crumpled up the square!
*
- from an engagement in the Mahdi wars where the unthinkable happened and a British Square gave.
Kinda but not exactly. The major purpose of forming a line is to maximize firepower. The fact that it makes cannon fire less deadly is a big bonus, but not the major gain. If possible, you rush cannon and take out the crew.
I agree with the point about corner vulnerability - but what do you mean by ‘stackability’ in this context? I assume that we’re not piling one infantry formation on top of another.
And if you’re refering to tiling a flat plane with the formation of your choice, then equilateral triangles (or isosceles right triangles) will tile the plane just fine, with the right arrangements. Actually, I suppose any right-angle triangles can, if only by forming pairs of them up into rectangles.
And on a more practical level, how would two equilateral triangle formations next to each other along one side not be defending each other? They’re even covering more of each other’s perimeter, proportionally, than squares.
Good point, and somehow very fitting with the username…
I think the advantage of a square over a triangle is simplicity. If you are in a group of, say 100 men, its really easy for everybody to get in a square shape- ten rows of ten; consequently, to stay in that shape you generally just have to make sure you have a guy on all 4 sides of you (unless you are on the edges).
But a triangle gets complicated. It would take a lot of discipline to form, and even then I don’t see any marked advantage. I’m guesing the complexity of the formation would also give it a disadvantage if infantry needed to form up quickly or rally from being dispersed.
The triangle were used for attack though; a tactic applied by the Vikings – a formation known as a fylking or svinfylking. I the first line there were two guys, in the second, three, then five, seven, and so on. A rushing fylking could then break the enemy line. As a side note, during the Viking era, it was believed that Odin (Allfather) himself invented the tactic.
There’s a not too informative Wiki article on the subject.
It’s the Navy that uses the Pythagorean Theorem for making leeway.
Less technology and more drill and discipline. The battle of Poitiers/Tours was won in just this fashion. The Arab army was probably outnumbered ( the most reliably detailed report of the battle, a Christian cleric subject of the Spanish Muslims, indicates this and under the circumstances it makes sense ) and after several days of maneuver failed to improve their position, their commander decided to risk opening the battle with a do or die frontal assault of his cavalry against the Frankish phalanx ( it should be noted that Arab armies of the day still tended more heavily to infantry than cavalry ).
The Franks of that time fought more or less “Roman-style”, in a tight phalanx, relying primarily on round shields and a gladius. No long spears or pikes ( though they certainly had javelins and other missile weapons ) and in general the Arab army would have better equipped, Frankish military manufacturing at the time being distinctly inferior. Nonetheless in the close combat that ensued the Arabs were badly mauled. In this case the difference wasn’t technology ( which was probably worse overall ), but just the advantage of a stong defensive position and highly disciplined troops.
This as opposed to other medieval encounters where poorly prepared levied infantry just broke when engaged by armored horsemen. Mighty scary having cavalry charge you - unless properly trained, the tendency is probably not to trust to your itty-bitty short sword made of iron or crap steel, like the Franks did under Charles Martel.
They do have a distaste for stepping on squishy objects. When jockeys take a tumble they ball up to tuck their limbs in, then pray they aren’t accidently kicked. Horses can be trained to overcome that instinct, though. Look up the origin of the term “ride roughshod” sometime.
Another advantage of the square was density. When firing in line, infantry would be in one or two ranks to maximize firepower. But when formed into square, the sides would be four or five ranks deep. This means that each charging horseman now has to fight many more opponents. The rule of thumb is that one cavalry man takes up the same frontage as three infantry. Charging a two-rank line puts one horseman against six bayonets or pikes. In square, each horseman will be facing twelve or fifteen pikes. And the full number of horsemen will not likely be able to hit the square head on, but have to wrap around the edges. This sort of blunts the inertia of the charge for the wrap-around cavalry. The four or five ranks also give the infantry a stiffer wall against the cavalry, physically and mentally.
The down side is that increasing your depth makes you a sweet target for the cannons. One canon ball can take your guys out like bowling pins.
and also Sir Henry Newbolt, Vitaï Lampada:
I think they were in line rather than a square, but I recall reading that one way to deal with cannon was to him the men lie down.
or to stay just below and behind the rise of a hill, waiting for the command to go up and over.
Yeah, but with the threat of a cavalry charge, the men would have to stay up. A good general could use the threat of cavalry to force the enemy infantry to form square just so that he could then annhilate them with artillery. The speed at which infantry could form square from the line and back again was very important for this reason and shows why textbook marching maneuvers were so thoroughly drilled into the men.
Northern Piper, reverse slope positions were credited for much of Wellingtons success at Waterloo.
Man, that’s cold.