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## Math for fun and glory

### Course: Math for fun and glory>Unit 1

Lesson 9: Other cool stuff

# 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.

## Want to join the conversation?

• Hi Vi, how can one fold a paper into three equally long parts
• 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.
• 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?
• 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.
• At , how does step "0" exist??
• It's not really a step, you can start at step 1 if you have a square piece of paper. It's a common joke if you don't need to do step 1
• 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.
• Vi supports Tau and dislikes Pi, so I doubt it was intentional. It is a funny coincidence!
• Wow, that's some amazing origami. I would probably fail within 2 seconds if I even attempted that. Oof :P
• How many videos have you done?
• Vi Hart has done about 50 videos on Khan Academy and counting, and I'm not sure, but there may be more on YouTube.
• 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.
• 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.
• 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)
• c is always the hypotenuse and a and b are the legs of the triangle. Also c2-b2=a2 and c2-a2=b2. It really doesn`t matter what you label a and b as, as long as c is the hypotenuse.