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Origami proof of the Pythagorean theorem

Leave your homework in the comments. Extra points for clarity and conciseness! Special thanks to my peeps at NYU where the idea for this video popped up during discussion. Created by Vi Hart.

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  • leaf green style avatar for user -.-
    Hi Vi, how can one fold a paper into three equally long parts
    Always wondered about that one
    (104 votes)
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    • leaf green style avatar for user Daryl
      There's one neat way to fold a paper into thirds. It comes from the series:
      1/2 - 1/4 + 1/8 - 1/16 + 1/32 - ... = 1/3
      That is, if you take the alternating sum of fractions where each is half the previous one, starting with 1/2, the sum approaches (After infinitely many terms) the value 1/3. You can do this out on paper:

      Make a fold at the halfway point of the paper. This represents 1/2. Then fold the left half of the paper in half. That represents subtracting half of 1/2, or 1/4. Next, add 1/8: Fold the paper in between the previous two folds. Continue that pattern, always folding between the last two folds you made. Eventually you'll get to where the last two folds are too close together to do much more, and that point is 1/3 of the paper.

      Then fold the right half of the paper to the 1/3 point to make the 2/3 crease, and you've folded the paper into three equally long parts. If you don't feel like making infinitely many folds along the way, you can also just mark the 1/2, 1/4, etc, points with pencil.
      (107 votes)
  • old spice man green style avatar for user NotaNewb117_664
    in any perspective, you can clearly see that near the end of the video she pointed out it could be a dolphin, my question is: if perspective is based on the person/persons that imply it, than is there a right or wrong in a general perspective? or more clearly, why would you have to describe the attributes of a dolphin, if in your perspective, it could be anything? who should i believe?
    (13 votes)
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    • leaf green style avatar for user Matthew John Ting
      Hrm, I think she's speaking about defining things like a mathematician, not about arbitrary perspective. At she says "...in which case you should define what a dolphin is and show that this fits that definition." So, if we define a dolphin as a parallelogram with 4 equal sides and 4 right angles that can do little wobbly flappy thingies when you shake it, then yes, this is a dolphin.
      In mathematics we try not to imply arbitrary things, we define the terms we are using and if we take certain things as given we state that clearly before we start the conversation. So, as long we all understand we're all talking about the same four sided dolphins, we're fine.

      You bring up an interesting point on belief. One of the things about math is that you don't have to "believe" others to come to a truth. Given the same starting points, you can follow logic to get there on your own without having to take the word of anyone else on it.
      What you have to do beforehand is agree on the starting points, but this still doesn't necessitate belief, just an agreement beforehand of what the parameters for this particular discussion will be. Those parameters can change, like in non Euclidean geometry.
      (15 votes)
  • duskpin ultimate style avatar for user Nicholas Ngo
    At , how does step "0" exist??
    (6 votes)
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  • hopper happy style avatar for user GCluff
    I can't help but notice that the time of this video is (pi). Did Vi Hart do this on purpose? Does anyone else notice this? Thanks in advance.
    (4 votes)
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  • starky ultimate style avatar for user Vidit K. [Mostly Inactive]
    Wow, that's some amazing origami. I would probably fail within 2 seconds if I even attempted that. Oof :P
    (4 votes)
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  • aqualine ultimate style avatar for user Evelina Marie Rivas
    How many videos have you done?
    (3 votes)
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  • female robot grace style avatar for user Alexandrea (Alex) Wright
    I just noticed is the length of this video..Pi much? Actually, Vi supports Tau, and hates Pi, so I don't see why this would be on purpose.
    (2 votes)
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    • duskpin ultimate style avatar for user KTKF
      Eh... She doesn't so much as hate pi as actively support the movement to make people understand it's not perfect, nor is it even the most useful of the circle constants. So, it's possible she simply put it their because it's math-y, and she wanted to.
      (2 votes)
  • female robot ada style avatar for user Imani
    i know to figure this out is a2+b2=c2, but how are you suppose to know which side is a,b, or c? if you guys know, am i suppose to know a bunch of stuff, because i'm only in 6th grade. do you have to do the sin,cosin, and tan( cosecent, something else, cotangent.) because i already know how to decifer which side is the hypothenues, oppisite, and ajacent. also, how do you figure out a 30 degree angle from a 60 degree. maybe i should go on sai's video?
    (1 vote)
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  • piceratops ultimate style avatar for user gunmayhanda
    Isn't this just Bhaskara's proof but with paper?
    (3 votes)
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  • aqualine ultimate style avatar for user Yasada De Silva
    How is this the pythagoras theory?
    (2 votes)
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Video transcript

You don't need numbers or fancy equations to prove the Pythagorean Theorem, all you need is a piece of paper. There is a ton of ways to prove it, and people are inventing new ones all the time, but I am going to show you my favorite. Only instead of looking at diagrams, we're gonna fold it. First, you need a square, which you can probably obtain from a rectangle if you ask nicely. Step one, fold your square in half one way, then the other way, then across the diagonal. No need to make these creases sharp, we're just taking advantage of the symetries of the square for the next step. But, be precise. Step 2: Make a crease along this triangle, parallel to the side of the triangle that has the edges of the paper. You can make it anywhere you want. This is where you are choosing how long and pointy, or short and fat, your right triangle is going to be, because this is a general proof. Now when you unwrap it, you'll have a square centered in your square. Extend those creases and make them sharp, and now we've got we've got four lines all the same distance from the edges, which will allow us to make a bunch of right triangles that are all exactly the same. Step three: fold from this point to this one. Basically taking a diagonal of this rectangle. Now we've got our first right triangle. Which has the same shape and area as this one. Let's call the sides: "A little leg", "a big leg", and "hypothenus". Rotate ninety degrees, and fold back another triangle, which of course is just like the first. Repeat on the following two sides. The original paper minus those four triangles, gives us a lovely square. How much paper is this ? Well, the length of a side is the hypotenus of one of these triangles. So the area is the hypotenus squared. Step four: unfold, and this time let's choose a different four triangles to fold back. Rip along one little leg, and fold back these two triangles. Then you can fold back another two over here. The area of the unfolded paper, minus four triangles, must be the same, no matter which four triangles you remove. So let's see what we've got. We can divide this into two squares, This one has sides the length of the little leg of the triangle. And this one has sides as long as the big leg. So the area of both together, is little leg squared, plus big leg squared. Which has to be equal to this area, which is hypotenus squared. If you called the sides of your triangle something more abstract, like: a, b, and c, you'd of course have a squared plus b squared equals c squared So quick review: Step 0: Aquire a paper square. OK, Step One: Fold it in half three times. Step 2: Fold parallel to the edges anywhere you choose and extend the crease. Step Three: Fold back four right triangles around the square and admire the area hypothenus squared that is left over. Step Four: unfold and rip along a short side to fold back another four right triangles and admire the area one leg squared plus the other leg squared that is left. And that is all there is to it! Of course, mathematicians are rebels and never believe anything anyone tells them unless they can prove it for themselves. So be sure to not believe me when I tell you things like: This is a square. Think of a few ways you could convince yourself that no matter what the triangles on the outside look like, this will always be a square, and not some kind of a rombus or parallelogram or dolphin or something. Or, you know, maybe it is a dolphin, in which case you should define what a dolphin is and then show that this fits that definition. Also, these edges look like they line up together. Do they always do that? Is it exact?