A trip around the world!

The earth is round.

Let's call a "trip" a clockwise revolution around earth.

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(1)

First, three planes start from the air base. At $1/8$ of the trip, all planes have $\dfrac{3}{4}$ of their fuel.

One plane gives the other two $\dfrac{1}{4}$ respectively, and returns to the air base with its fuel $\dfrac{1}{4}$ left.

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(2)

Then two planes continue going. At $1/4$ of the trip, both of them have $\dfrac{3}{4}$ of their fuel.

One plane gives the another $\dfrac{1}{4}$ , and returns to the air base with its fuel $\dfrac{2}{4}$ left.

Then one plane is left, with its fuel full.

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(3)

At $1/2$ of the trip, the plane's got $\dfrac{2}{4}$ of its fuel left.

Then the trick happens. One plane starts off at the air base, going to the opposite direction where the first three airplanes went.

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(4)

The two planes would meet at $3/4$ of the trip, and one plane would have its fuel empty, and the other, $\dfrac{2}{4}$ .

The two planes then share the fuel fairly so that both of them would have $\dfrac{1}{4}$ each.

At the moment that happens, another plane starts off at the air base, going to the direction of the fourth airplane.

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(5)

The three planes would meet at $7/8$ of the trip, and two of them would have its fuel empty, and the other, $\dfrac{3}{4}$ .

The three planes then share the fuel fairly so that each of them would have $\dfrac{1}{4}$ .

The three planes return to the air base, and at the moment they arrive, all of their fuel tanks gets empty.

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Therefore we need $5$ planes to do this, and it is impossible to do it with less planes.