Was it not possible to exploit the gaps between each pre-modern melee formation?

Soldiers do not like to be surrounded. This simple fact drives nearly all battle tactics, ancient or modern, ground or air. A soldiers' eyes, weapons and protection all point forward. Harass a soldier's flank, and they will retreat. The most consistently successful ancient formations and tactics were all about addressing this fact, along with the difficulties of commanding an army with nothing more than flags and runners. Break the enemy's formation and you win.

Attacking a gap in the line of an ancient formation with fast moving forces may bring you victory. The goal is to pass through quickly, before the enemy heavy infantry have time to react, and fall upon their rear. This is why ancient formations were relatively compact, even between their companies, and most cavalry battles took place on the wings where there is room to maneuver.

Attacking a gap in the line with slow moving heavy infantry is a sucker's bet. Let's say the armies of Numbers are facing the armies of Letters. Numerically equivalent, but differently deployed.

    111        ZZZ
    111        ZZZ
    111
               YYY
               YYY
    222
    222        XXX
    222        XXX

What happens when Y chooses to march into the gap between 1 and 2? Y's weapons are facing forward, neither at 1 nor 2. Their flanks are exposed. If they turn to face either formation, their rear will be exposed to the other. This is a terrible position.

To make matters worse, both sides have skirmishers, cavalry and reserves available. As Y marches into the gap between 1 and 2, General Number will be ordering reserves to slow Y's forward progress preventing them from punching through the line completely and attacking from the rear.

Finally, the rear ranks of 1 and 2 can sidestep to extend the line and close the gap.

     11ZZZ
    111ZZZ
    111
    1YYY
    2YYY
    222
    222XXX
     22XXX

Y is now surrounded on 3 sides. The further they push at the weak center, the worse they are surrounded. Because of the strong tendency of ancient formations to press forward from the rear. Unless X and Z do something, Y will rout.

     11ZZZ
    111ZZZ
    111
    1            YYY
    2            YYY
    222
    222XXX
     22XXX

With Y routed, there is now a huge hole in Letters' line. Cavalry, skirmishers, reserves and the flanks of 1 and 2 pour in, turning the flanks of Z and X. X and Z can no longer support each other. While X is held in place, Z is surrounded and routed. Numbers then descends on X.

This is similar to what happened at the Battle Of Cannae when Hannibal defeated a far larger and equally well trained Roman army.

The Battle of Gaugamela saw multiple penetrations of the Macedonian line by cavalry. A charge by chariots was thwarted by deliberately allowing them to pass through the infantry, only to be defeated by reserves.

Late in the battle, Persian cavalry successfully navigated a gap in the Macedonian line. Had they exploited it by falling upon the failing Macedonian left, the battle may have been won. Instead they sacked the Persian camp and were turned away by reserves. This failure shows the importance of discipline in ancient warfare; give a bunch of amateurs the choice been fighting and looting and they will loot.

A similar tactic plays out in many ancient battles. The center is made deliberately weak with orders to slowly withdraw when pressed. An overconfident attacker presses forward and is destroyed by strong forces pressing their flanks.

This plays out even in modern warfare. During Operation Cobra, the Allied breakout from the Normandy beachhead, the Germany army, on the verge of being surrounded, attacks the apparently weak American center hoping to cut off the American spearhead. In reality, they make little headway. The time wasted on the counterattack only guarantees the German army in the west is trapped and nearly destroyed.

Alternatively, attacking the gap can result in victory. Maybe the gap between 1 and 2 is wide enough for Y to maneuver. Maybe General Number is slow to react. Maybe the General reacts quickly, but communications are poor and the orders don't reach 1 and 2 fast enough. Maybe Number's cavalry has been defeated, or is off doing something dashing, and Number's reserves have already been committed.

Following behind a screen of skirmishers, Y can split and attack 1 and 2's flanks simultaneously. This maneuver on Y's part requires great training and discipline. The soldiers of each wing of Y must raise their weapons vertically in the air, rotate 90 degrees as a unit, lower their weapons and attack in the new direction, all while being harassed by the enemy. X and Z thin their ranks to extend their lines to prevent having their own flanks turned.

    111ZZ
    111ZZ
    111ZZ
    YYY

    YYY
    222XX
    222XX
    222XX

Numbers is in trouble. 1 and 2 have been separated and can no longer mutually support. Each are being attacked on two sides and are in danger of being surrounded, while X, Y and Z's weapons all face forward.

In modern warfare, the enlarged scale offers many more opportunities to exploit boundaries. The lines between strategic formations are often exploited. Many times in World War I and II, an army or national boundary was chosen as the target of an offensive. The, often correct, assumption being that communication and coordination between armies and allied nations will be poor. Because of the hierarchical nature of armies, the defending units on either side of the boundary may not be able to communicate with each other. They may not even know where their counterpart is located. The defending units may not have maps or transport beyond their own areas of operation, hampering their ability to support their neighbors. They may not speak the same language, use the same radio frequencies, equipment, ammunition, spare parts or fuel. They may not be able to call in artillery support nor air support from their neighbors.

In modern warfare, the greatly enhanced speed, firepower and control of units makes exploiting a gap a tactical possibility... though you still don't want to get bogged down in the middle. As the Thunder Runs of the Battle Of Baghdad it is a great risk and the penetrating unit can only survive so long without support. Even in the 21st century, soldiers do not like to be surrounded.