[EDIT: I got "lines of longitude" mixed up with "lines of latitude"...all better now!]
There are two equinoxes per year - on March 20th and September 22nd...give or take a day because of leap years.
All places on the same line of longitude see the sun at it's highest point at noon on that day (unless they are in some crazy time-zone screwup).
At that moment, the sun is due north or south of the zenith of the sky depending on whether you are north or south of the equator.
This means that lines of longitude in FET must be straight north/south lines.
That disproves the "bipolar" map because it has curved lines of longitude. On the equinox, some places on Earth would see the sun somewhat to the east or west of the zenith.
That leaves us with a unipolar map as the ONLY possible FE map. It could be either the current (discredited) unipolar map - or it could be one with the south pole in the middle and an "ice wall" around the north pole.
Sadly, we know that the unipolar map can't be true because the Southern cross constellation can't be south of every place in the southern hemisphere at once...and a similar argument would apply with the "inverse unipolar" map and Polaris.
Hence the unipolar map can't be true either.
Since no bipolar map can have straight lines of longitude - and there are only two possible unipolar maps, neither of which work...it can only be that the world isn't flat.
(Probably Tom will tell us that magic perspective fixes this - light rays tying themselves into pretzels, yadda, yadda.)