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
> But hasn’t your body done a 360?
The interior angles of the triangle are not the same as the exterior angles. You are not turning 60 degrees left to follow each leg of the triangle, you are turning 120 degrees. Turning 120 degrees three times *is* 360 degrees.
60 degrees is the inside measurement. In your example you are facing the outside, not the inside, so you want to turn 120 degrees. 3 * 120 = 360!
It has, but your body turning isn’t what defines the inside angles. That’s not what your body turning was measuring.
Did anyone else try to visualize this and end up on the opposite side of half a hexagon facing backwards?
The only way the instructions would make sense to me as going around a triangle would be:
1. Walk forwards 1 mile in a straight line
2. Turn 60° to the right
3. Walk backwards 1 mile in a straight line
4. Turn 60° to the right
5. Walk forwards 1 mile in a straight line
Draw it out on paper. The exercise is asking you to form a triangle, but the WORDING of it is tricky and your brain just automatically replaces the angles because we gloss over things that are wrong.
Draw a straight line and then draw the perpendicular, and 60 degrees is less than 90, so turn a little bit LESS than that perpendicular. Are you going to form a triangle going this way? Nope, you’re going to form a hexagon.
The fact is, you have to turn 120 degrees 3 times in order to draw a triangle. If you turn only 60 degrees you won’t form a triangle you’ll form a hexagon.
So the exercise should say: walk one mile, then turn 120 degrees then walk another mile and repeat. The exercise is tricking you with asking for 60 degree turns, and when you draw it on paper your brain just substitutes the angles of the triangle for the angles you turned. But you turn 120 degrees not 60.
And 3×120=360.
You are actually turning 120° each time
Take a stick and then align it with the edge of your triangle, and then rotate it around the corner, you turn it a full 120°
If you only make 60° turns, you end up making a hexagon
The exterior angles of a polygon always add to 360°
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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