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My favourite proof without words is the proof that the sum of the interior angles of any triangle is 180 degrees. Proof: draw an arbitrary triangle in chalk on

My favourite proof without words is the proof that the sum of the interior angles of any triangle is 180 degrees. Proof: draw an arbitrary triangle in chalk on the ground. Stand on one of the lines at one of the corners and note the direction you are facing. Now back up until you are at the corner of the triangle behind you. Rotate your self by shuffling inside the angle to the next line. Walk to the next corner, repeat. Walk (backwards this time) to the last corner, repeat. Now shuffle back to your starting position. You are facing exactly 180 degrees in the other direction! Of course it is way better to just do this than write it down! But obviously I can't do it on HN.

whatshisface 8y ago
Does that prove that every triangle has angles adding up to 180, or just that one?
- Corvus 8y ago
  “draw an arbitrary triangle” means this works with any triangle.
  - whatshisface 8y ago
    But you'd have to try an infinite number of triangles before you were convinced that it worked for all of them.
    - justinpombrio 8y ago
      This is an interesting confusion. OP said "draw an arbitrary triangle in chalk on the ground", and from there gave a set of instructions that showed you how to check that the interior angles of that triangle added up to 180 degrees. You're right that that concrete set of instructions only proves it for that one triangle. But that's not the whole proof. If I followed these instructions for a right triangle with angles 45,45,90, I'd only have proved that that triangle's angles added up to 180 degrees. The full proof is that if you imagine doing this, for any triangle, it's clear that it will always work. When I picture this in my head, it's clear that I'm going to end up having turned 180 degrees, regardless of the measurements of the triangle. This leap: starting with a set of concrete instructions that you can do on a particular object, but then verifying that they would work no matter what object you started with, is common in proofs. EDIT: Nevermind, totally unrelated.
      - whatshisface 8y ago
        Why is it clear that it will always work? It would be a proof if that part was written down. Since we all already know that the angles add up to 180, it might just be the intuition taught to us by another proof of the statement leaking backwards into the implicit step in this attempt at proving the statement. The part that's left off (the proof that it always works) is actually the bigger part of the contraption.
        justinpombrio 8y ago
        Because you end up back on the line you started at, facing in the opposite direction. That's a 180 degree turn.
        whatshisface 8y ago
        Suppose I found a triangle whose internal angles added up to 190 degrees. If I did the experiment on it, I would end up 10 degrees away from where I was predicted to be. How can this scenario be ruled out?
        justinpombrio 8y ago
        Both of these things are true: (i) Like you said, if you found a triangle whose internal angles added up to 190 degrees, and you followed the procedure, you would have turned 190 degrees, rather than 180 degrees. This is true because during the procedure, you turned three times, each by one of the angles of the triangle, so the total amount you turned was the sum of the angles. (ii) You would end up back at the line you started with, facing in exactly the opposite direction. This is true because the last step of the instructions is to turn until you're facing in the direction of this line. Thus you have turned 180 degrees. Now of course this is nonsense: you can't have turned 180 degrees, but also turned 190 degrees. How did we arrive at nonsense like this? The logic is sound, so it must have been one of the assumptions. Which assumption is questionable? Oh, right, the triangle whose angles added up to 190 degrees. This is a proof by contradiction, that shows that a triangle whose angles add up to 190 degrees cannot exist.

whatshisface 8y ago

Does that prove that every triangle has angles adding up to 180, or just that one?

Corvus 8y ago
“draw an arbitrary triangle” means this works with any triangle.
- whatshisface 8y ago
  But you'd have to try an infinite number of triangles before you were convinced that it worked for all of them.
  - justinpombrio 8y ago
    This is an interesting confusion. OP said "draw an arbitrary triangle in chalk on the ground", and from there gave a set of instructions that showed you how to check that the interior angles of that triangle added up to 180 degrees. You're right that that concrete set of instructions only proves it for that one triangle. But that's not the whole proof. If I followed these instructions for a right triangle with angles 45,45,90, I'd only have proved that that triangle's angles added up to 180 degrees. The full proof is that if you imagine doing this, for any triangle, it's clear that it will always work. When I picture this in my head, it's clear that I'm going to end up having turned 180 degrees, regardless of the measurements of the triangle. This leap: starting with a set of concrete instructions that you can do on a particular object, but then verifying that they would work no matter what object you started with, is common in proofs. EDIT: Nevermind, totally unrelated.
    - whatshisface 8y ago
      Why is it clear that it will always work? It would be a proof if that part was written down. Since we all already know that the angles add up to 180, it might just be the intuition taught to us by another proof of the statement leaking backwards into the implicit step in this attempt at proving the statement. The part that's left off (the proof that it always works) is actually the bigger part of the contraption.
      - justinpombrio 8y ago
        Because you end up back on the line you started at, facing in the opposite direction. That's a 180 degree turn.
        whatshisface 8y ago
        Suppose I found a triangle whose internal angles added up to 190 degrees. If I did the experiment on it, I would end up 10 degrees away from where I was predicted to be. How can this scenario be ruled out?
        justinpombrio 8y ago
        Both of these things are true: (i) Like you said, if you found a triangle whose internal angles added up to 190 degrees, and you followed the procedure, you would have turned 190 degrees, rather than 180 degrees. This is true because during the procedure, you turned three times, each by one of the angles of the triangle, so the total amount you turned was the sum of the angles. (ii) You would end up back at the line you started with, facing in exactly the opposite direction. This is true because the last step of the instructions is to turn until you're facing in the direction of this line. Thus you have turned 180 degrees. Now of course this is nonsense: you can't have turned 180 degrees, but also turned 190 degrees. How did we arrive at nonsense like this? The logic is sound, so it must have been one of the assumptions. Which assumption is questionable? Oh, right, the triangle whose angles added up to 190 degrees. This is a proof by contradiction, that shows that a triangle whose angles add up to 190 degrees cannot exist.