We've basically seen two efforts at flat earth maps here.
1) The one in the Wiki has the "North Pole" in the center with straight lines of longitude radiating away from it until they reach the "ice wall" surrounding the habitable parts of the Earth...and concentric circles for the lines of latitude.
2) The one that Tom occasionally posts that has both the north pole
and Antarctica present, with wavy lines for latitude and longitude.
Now we know that Tom currently denies the first map (the one on the left, above) because he knows that the continent of Antarctica actually exists. He also says that he doesn't know the "true" FET map of the Earth...which I suppose is fair enough.
However, there is a problem with the map on the right.
The star "Polaris" is always in the North...vertically above the North Pole. This is a fact that has been relied upon for celestial navigation for hundreds (probably thousands) of years.
But if the map on the right is correct, then Polaris would NOT reliably point "North" in as much as the lines of longitude are curved and waving about all over the place.
Tom tells us that he believes that light travels in straight lines - so ANY FE map that has curved longitudinal lines is demonstrably wrong.
This severely restricts the range of possible maps that can be considered. Any FET map that allows common celestial navigation has to have straight lines of longitude in any region where Polaris is visible (ie the entire region North of the Equator...PERIOD.
FINDING #1: Lines of longitude in the Northern hemi-plane must be STRAIGHT.
Unfortunately, there is no similar star close to the South pole...but sailors have still managed to navigate there using the "Southern Cross" method. This involves a grouping of stars which allow you to infer the location of the South pole from the intersection of imaginary lines:
So although we don't have a convenient "anti-polaris" to work with - we can still use the stars to indicate a position that must be vertically above the South Pole.
(UPDATE: With a moderately good telescope, you can use the star "Sigma Octantis" as a "southern-pole-star because it is sufficiently close to being vertically above the South Pole)
OK - so using the same argument as with Polaris - to keep the Southern Cross (and Sigma Octantis) directly in the South - the lines of longitude must be straight there too - and that's a big problem for FET.
If you subscribe to the map on the left - the Southern cross would have to be above every point on the Ice Wall at once. All of those places are "south" when viewed from somewhere. Someone living in South Africa would see the Southern cross in their southern sky - and so would someone in Australia....and that's impossible.
So the map on the left is busted too.
FINDING #2: Lines of longitude in the Southern hemi-plane must be STRAIGHT.
What FET needs is a map where all of the lines of longitude run straight from the equator to the North Pole and also straight from the equator to...wherever the heck the South Pole is.
But now...unavoidably...all of the lines of longitude must have a 'kink' in them at the equator...which is truly batshit crazy...ships and aircraft would have to make sharp turns as they crossed the equator in order to keep on a southerly course.
What would happen to compasses? How could they possibly agree with celestial navigation?
It's actually even worse than that...unless the equator is a dead straight line, there would be no way to avoid some lines of longitude from crossing each other! But if the equator is a straight line, then circumnavigation becomes impossible (note this is a problem for the map on the right).
There is no possible Flat Earth mechanism to keep both Polaris and the Southern Cross in the right places for both celestial navigation and compass navigation.
FINDING #3: There is no possible "flat" map that can explain both celestial and compass navigation...and since this has been in use for many hundreds of years (perhaps thousands) - we know that the Earth must be round.
QED.
(I'm betting Tom will be ignoring this one!)