Animating the Mercator Projection

:)

most countries of the southern hemisphere are in equatorial regions which are not destorted.
Exept for Argentina and Australia/NZ in a small amount- You can see it in the gif if you look closely.
There'd be a huge distortion for Antarctica though, but it's missing in the gif

Try to peel an orange and end up with one straight contiguous peel at the end.
No way to do so. Even if you manage to minimize the cuts- peel will burst as soon as you try to flatten it.
Neither can it be done with an earth map.
One simple way to get a square earth map is to project the whole map onto a cylindric shape enveloped around the globe which then can easily be cut open and enrolled.
Obvious drawback:
equatorial countries are depicted much smaller than those beyond tropics.