Around the world assuming that all of the aircraft must return safely to the airport?

Question

On Bagshot Island, there is an airport. The airport is the home base of an unlimited number of identical airplanes. Each airplane has a fuel capacity to allow it to fly exactly 1/2 way around the world, along a great circle. The planes have the ability to refuel in flight without loss of speed or spillage of fuel. Though the fuel is unlimited, the island is the only source of fuel.
What is the fewest number of aircraft necessary to get one plane all the way around the world assuming that all of the aircraft must return safely to the airport? How did you get to your answer?

Notes:

(1) Ignore extra fuel consumption as a result of acceleration, evaporation of fuel, bleeding-heart-liberal fiscal policies, etc.
(2) All the planes have to make it back safely, so you can’t give all your fuel away to another plane.
(3) Assume that refueling is an extremely fast process.

Answer 1

It will take seven planes to do this.

1st plane will be looping from Base to 1/6th point and back so at the 1/6th point it will have 1/3rd fuel to give each time. Let's call this point A.

Now similarly 2 (ie., 2nd and 3rd) planes can form such loop at the 1/6th the and 1/3rd points with the 3rd plane looping such that it meets with the 2nd plane refuel there and goes to 1/3rd point and is able to spare 1/3rd fuel. Let this be point B

Just like this the 4th, 5th and the 6th can form a triple loop that spans half the distance where the 6th will also be able to spare 1/3rd fuel at the end. Let this be point C.

Now the 7th plane will be able to refuel at the points A,B & C and have enough fuel for a round trip.

Please note here that the planes that form the loops can also refuel at points A and B to be able to spare fuel and once they are not needed all the planes starting from the farest will have enough fuel to get back to the airport.