I’m having trouble wrapping my head around this one:
You start from a given set point, and walk one mile in a straight line, then turn 60 degrees to the left. You then walk another mile in a straight line, and then turn 60 degrees to the left. Finally, you walk another mile, and finish with a 60 degree turn to the left. You are now back at your start point, facing the way you started. Then angles add up to a triangle, 180 degrees. But hasn’t your body done a 360?
In: Mathematics
What you are describing has a generalization known as the turning tangent theorem btw. The amount the tangent turns on a closed piecewise differentiable curve (it has smooth intervals + some vertices where the derivative jumps) + the exterior angles is 2π. Since the lines in your case are straight, the angle of your tangent is 0 so your exterior angles mus already be 2π. It seems fairly simple but is actually a little tricky to prove.
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