In a fight you want to maintain superiority. If you are three airplanes fighting against two enemies you are likely going to win. But if your three airplanes are spread out over a larger area then the two enemies might attack one and one airplane winning all three fights. So you want to stick together so that you can fight together and defend each other. The easiest way to do this is to fly tight enough that you always see each other and can follow each others exact move. If you get too far away from each other it is easy to lose track of each other, miss a turn and get separated.
In addition to this it helps when scouting for enemies. You can have one looking forward, one to the left and one to the right. When you focus on just one section of the sky you are more likely to spot anything. This applies both when looking with your eyes and when focusing your radar on that section.
And when you have a lot of airplanes flying it can become too much for a controller and other pilots to keep track. So instead of keeping track of each individual airplane you keep track of each formation. It is easier to keep track of 10 formations then 30-50 single airplanes. You can even do parallel take-off and landings to increase the capacity of the airfield as the airplanes can fly in formation through these operations as well.
But the tight formations you might see in events and on film is mostly just for show. They do this for training, and there are certain situations when you want to get this close during a combat mission. But most of the time they fly a bit further from each other so they have some room to move around in case of mistakes or turbulence. You are typically not touching wingtips.
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