Drawing three right angles on a sphere - GIF

[mklotz]

Well, as it turns out there are an infinite number of points where a walk according to my description is possible.

The north pole is obviously one but, TM51, the south pole isn't because you can't walk south from the south pole.

Now consider this...

Draw a circle of 1 mile CIRCUMFERENCE (radius1 = 1/(2*pi)) around the south pole. Call this circle 1

Draw another circle around the pole of radius2 = 1 + radius1. This circle will lie one mile north of circle 1. Call this circle 2.

Stand on any point on circle 2 and walk a mile south. This will put you on circle1. Walk a mile east and you'll traverse circle 1 and return to the point where you entered circle 1. Now walk a mile north and you'll return to the place you started on circle 2.

There are an infinite number of starting points on circle 2 so there are an infinite number of points where my described tour is possible.

Yes, geometry is fun (and sometimes subtle).