Near the South Pole, there are latitudes where the circumference is 1 mile, 1/2 mile, 1/3 mile... You are 1 mile north of those latitudes.

ok, so haversine distance is a thing on a sphere: http://en.wikipedia.org/wiki/Great-circle_distance

You are a bit over a mile from the south pole (how much a of bit requires math i don't feel like doing, but is very important) You walk towards the pole, turn and walk west in an exactly 1 mile circumference circle around the pole (this is why the "bit" was important) and you walk back north to where you started.

There is only one valid point on the northern hemisphere (the north pole) and a ring of points at the south pole.

found the math I need! https://www.astro.virginia.edu/class/whittle/astr553/Topic16...

C = R sin(r/R) R = radius of earth aprox( 3,959) C = 1 mile solution is r+1 distance from the south pole

Aren't there multiple rings for the South Pole solution? There's the solution where you head south, circle the pole once, and head back north; then there's also the one (starting slightly further south) where you head south, circle the pole twice and go back... and so on.

yep! I missed that, infinitely many concentric rings.