The world is spherical not square, which means that the farther away from the equator you are the smaller a circle parallel to the equator is.
If you tried to use a map of the northern hemisphere for the southern hemisphere without turning it upside down, you would be stretching the point that is the north pole to cover the entire circle that is the equator, and condensing the entire circle that is the equator to the point that is the north pole.
If you tried to use a map of the northern hemisphere for the southern hemisphere without turning it upside down, you would be stretching the point that is the north pole to cover the entire circle that is the equator, and condensing the entire circle that is the equator to the point that is the north pole.