Cec, with the South pole as point of axis draw a circle around it of one mile circumference, now start one mile north of the circumference of the circle.
Now you can walk one mile south until you reach the circumference of the circle, you then walk around the circle, a one mile walk, bringing you back to the point you reached when you walked the one mile south, you then retrace the one mile back north to where you started from.
You can have a circle of 1/2 mile circumference, you then need to walk around twice before you start to head north.
For my riddle it's the word "start" and an alternate meaning.
Start: make a sudden movement from surprise, pain, etc.
Polar bears come from the Arctic so our intrepid explorer coming across one in the Antartic caused him to start.
MCC