How to Trisect an Angle with Origami – Numberphile

How to Trisect an Angle with Origami – Numberphile

Well Euclid had the straight edge and the compass that those are the tools that he had available. He didn’t have very
good paper. A completely different approach that you can take to construction which is well it’s much less classical and so it’s less talked about is to use folding paper, instead of a straight edge and a compass Now, if you think about it, folding paper uses nothing. First of all, you can’t ever do circles. You’ll never be able to fold a circle onto a paper. So, all you have, ever, is straight lines. If you think about, is this stronger or weaker? than this set of tools that Euclid was using, then the intuitive answer is that it must be weaker, but sometimes the intuitive answer is not the right
answer. Let me just jump right in and show you how to trisect an angle by
folding paper. What i’d like you to realise is that essentially you
have a straight edge because if you have two points that you
previously constructed somehow then you can fold the paper so that those
two points are on the crease this is exactly the same thing as
connecting them by a straight edge. Even better you can do perpendicular
bisector in a single step because you can fold
the paper so that these two points cover each other and when you do that, that created crease will be perpendicular to the line
segment connecting those points and it will cut it exactly in half because they covered each other so to trisect an angle first we have to create an angle. A right angle is not very interesting You can trisect a
right angle with straight edge and compass which is probably an important point
to make. It’s not that there’s no angle whatsoever that you can trisect it’s that you can trisect an arbitrary angle
so ninety degrees you can trisect because you can construct a thirty degree angle. So we want to create an
arbitrary angle so that our job is not so easy so I’ll just fold the paper so that the bottom of this crease is right at that corner So this is the angle. We created it arbitrarily. We don’t know how big it is. So I’ll try to draw a line just inside the crease. Ok. Brady: you’re gonna trisect that?
Zsuzsanna Dancso: I am going to trisect that First we fold the paper in half so I just
fold the bottom up to the top so that they match exactly then I fold the bottom up to the half so now I have this crease at the
halfway so now I have these two creases one
up here and one down here Now, here’s the trick. I’ll mark two points one is this bottom of the angle Brady: Yep
ZD: the tip and the second one is this point on the edge of the paper that’s halfway up. I will fold the paper so that this point at the tip of the angle lines up/matches this line this crease that I created
before and this point halfway up the paper matches this line which is the angle
itself so I’m going to fold it and fiddle it around now once I’ve found that alignment, I’m going to put my hand down
and crease the line and then I take a pen and mark this point right where the tip got okay open it up and another important point to mark is
where this crease that I just created with this last fold where that intersects the bottom line so you might already see where this is going
Brady: yes ZD: I will fold to connect the tip of the angle with this marked point and then fold again to connect the tip of the angle with the other marked point and magic …
Brady: that’s it, is it? ZD: that’s it. Since we are folding we can check that, not prove it, but check it by folding the crease and see that it
matches up folding this crease and see that it matches up for the three so if you want to prove that …
it’s an exercise it’s doable if you remember your
congruent triangles Brady: What power does origami have that the straight edge and compass didn’t have?
ZD: exactly very good question. So the trick was this one step that I did which was taking these two points and lining them up
with two lines. So if you allow that step which is a reasonable step, I mean if I have a paper it’s very easy to do so the key is that if you translate this step to
what it does to coordinates the same way that I told you about it in
the context of straight edge and compass what it does is this crease that it creates is a shared tangent of two parabolas, and to solve this equation to find the shared tangent of two parabolas you need to do something cubic.
Brady: So that was the one thing that was beyond Euclid ZD: That was the one thing that’s beyond Euclid and it turns out that with origami phrased in terms of what are the constructible numbers what are all the coordinates that we can construct we
can do addition, subtraction, multiplication,
fractions square roots, and cube roots. Brady: So, origami is more powerful than Euclidean geometry!
ZD: It’s more powerful than Euclidean geometry. even better if I give you any cubic equation you can construct by origami the solution so you could solve a cubic equation
which is the formula is is really really big and ugly and hard to plug into and you can solve it just by a simple
folding mechanism and then measuring the solution so I for example I showed you how to do
square root two, I showed you how to do three I show you how to do one-third it turns
out that you can do all numbers that just involve fractions
and square roots and addition and subtraction but there’s
a problem with cube roots so what this guy proved is that you will never be able to do cube roots

100 thoughts on “How to Trisect an Angle with Origami – Numberphile

  1. Correct me if this is not a good/correct way to think about this, but it seems intuitive to me that, to solve cubic problems like these with as simple tools, one would need to move into the 3rd dimension as origami allows since one is folding the 2d plane in 3d's. So, using that logic, would one need to be abld to "fold" a cube in 4d space to construct things requiring variables to the 4th power?

  2. So without allowing the paper to bend through a higher dimension, third roots are impossible to construct? Would a forth dimensional being drawing a 3 dimensional schematic not be able to construct a 4th root without bending the 3d surface into the 4th dimension? Correct me if I'm mistaken, but it seems the whats limited is that you can only draw in 2-dimensions, so you can't used spheres to construct third roots the way you used circles to construct second roots.

  3. Very interesting video. I think this should be used much more in schools.
    On a side note, I LOVER Zsuzsanna's accent. It is beautiful!
    Thanks for another great video!

  4. Please upload a video showing everything which we can do using origami or tell me the link of similar video.

  5. Cool, makes sense that you have to preform an action in the third dimension to solve a cubed problem on a two dimensional medium.
    Though, that's coming from a layman in hindsight.

  6. If I told you it was possible, would you then be able to do it?
    Since you know it is possible, I mean? Pierre Wantzel was wrong. Ofcourse it is not impossible. Mathematics is proof of eternity, so why should there be a limitation?

  7. I think the video should be called "how to triplicate an angle using Origami". You're not trisecting; just triplicating. However, this is very interesting.

  8. Euclidean geometry uses straight lines and edges. It is indeed a 2 dimensional representation, but the tools we use i.e. constructing edges and angles are in 1 dimension. So, in general Euclidean geometry is a 2 dimensional representation of 1 dimensional operations. Hence it is limited to square roots. Whereas, in origami, we use folding of papers as the tool which involves 2 spatial dimensions marking a 1 dimensional crease. The entire representation is in 3 dimensions, i.e. the paper that we can hold. Hence, it can go up to cube roots. If we are able to make 3 dimensional operations in 4 dimensional representation, then measuring out fourth root is also possible. This can go on. This also proves that our brain is limited to map 3 dimensions, though infinitely many dimensions are possible.

  9. Correct me if I'm wrong, but by matching two points iteratively one only get's a close approximation of the actual points. What Euclide's line and compass can accomplish, in theory, are perfect lines, perfect points and perfect circles. So the end result is a perfect bisection for example. Using a ruler and a compass, one can create an approximation of a parabola, just as accurate as the origami matching points. So I have serious doubts this could be considered a proper solution. Especially that you mention that you only get straight lines by folding paper and in theory I could also get a pretty good approximation of a circle by folding paper into smaller and smaller triangles and the end result an n-gon will be a "good enough" approximation of a circle.

  10. What you really need is to be able to get parallel lines with a compass and ruler.

    And the tangent thing. But the tangent thing is WAY cooler. I just have personal history with the other.

  11. Just thought of a simpler way:
    1. Draw two lines forming any random angle..
    2. With a compass draw and cut a Circle on a separate paper..
    3. Place the center of the paper Circle over the angle Vertex, and mark the angle overture
    4. With the compass, draw and cut a smaller circle inside the original Circle, and make it be a Ring..
    5. Cut the Ring on the marked lines which formed the angle, leaving you with a Curved Stripe
    6. Fold the Curved Stripe, but not in half, fold it in thirds (as you may fold a regular non-curved paper)
    7. Unfold the Curved Strip and place it over the angle overture.. Ta-da!

    (I guess it could also be done with a tree leaf instead of paper, or perhaps even a string.. and by all means sorry if it's too obvious, i'm no mathematician)

  12. Couldn't you just create a program that took in equations in this manner and folded 3D geometry for you accordingly? You would then have the program measure what you needed, which is a very simple operation. This way thickness and human error would not affect the results, and you would also get very high precision values.

  13. Been over a month now… Have been chasing this problem .

    If anyone interested then check out the proof by Wantzel. Trisecting an angle using straightedge and a compass is same as solving cubics. Using multiple angle formula of a ( cos3 theta) = 4cos^3 theta – 3cos ( theta) you can tranform this problem into solving cubics.
    Clearly, if 60 degrees is trisected then each angle would be 20 degrees.

    You can show that using Wantzel method, cos(60 degrees) can not be solved and hence prove that arbitrary angle cannot be trisected.

  14. So you could solve 4roots and terracubic (I don't know the word…4D cube) with rectangular prisms having small measurements in the fourth dimension and folding them into it?

  15. What about fifth roots, seventh roots, eleventh roots? I'm guessing there cannot be a geometric way to solve those and perhaps it has something to do with not being able to solve polynomial equations past the power of four analytically? Or maybe Fermat's last theorem?

  16. I could be wrong but is origami using a sort of compass by sing the third dimension by the folding mechanism. If so then the problem involves the 3rd dimension. So in essence is able to solve cubic equations. Can this be generalized to higher dimensions and thus solving higher order equations?

  17. I have a question, I tried it myself and found out that it doesn't work, but I might have made some error, but why isn't it possible with the compas and straight edge? I tried to start like if I would disect the angle, than did the same with the compas with the side marks on the lines and the point on the disected line. (I don't know if that makes sense) I get three lines, one going through the middle, which isn't important (doesn't need to be drawn at all actually) and two lines left and right from the middle. I than meassured the angles and it doesn't fit. Can someone point out the issue I have? Thanks 😉

  18. "Origami is more powerful than Euclidian geometry" – well "Origami is more powerful than Euclidian construction" would be more exact. But we get the point.

  19. I thought in a previous video you showed out to trisect a line. Why couldn't you use that to trisect an angle?

  20. cant u just take the width of the angle at like 10 cm in, copy it over to a small strip of paper, so you know how wide the angle is at a certain point, that strip of paper you can easily fold into 3 equal parts, place that strip back at that 10cm marker, and voila, you have EXACTLY devided the angle into 3 equal parts lol

  21. Years ago the American "60 Minutes" TV show had a segment about a young math prodigy "Jacob" whose clever mental shortcuts for quadratic equations entailed a system of triangles & other polygons. Any thoughts on that +Numberphile?

  22. So if you can trisect by origamy it is logical that you can by euclid as my son and I did. I think the proof was wrong because it was not geometric and the universe is geometric and not cartesian.

  23. Seems like there would have to be some way to use the trisectable 90-degree angle and some sort of projection method, to trisect an arbitrary angle.

  24. is not origami like fancy geometry? why not include origami in straight edge constructions to allow things that weren't allowed before

  25. it seems natural that with 3 tools you can construct equations of power 3. i wonder if there is some fourth tool that allows 5-sect and 5-root? or is that possible with the 3 given tools?

  26. awesome… in the other video about the compass and ruler, I thought about the solution to trisect an angle and did start to think about a three-dimensional solution. Thanks! you confirmed my onset of an intuition (although I haven´t done it!).

  27. What about with an arbitrary shaped piece of paper? Using paper with 4 right angles and parallel sides without building those features seems unfair.

  28. Drawing circles and straight lines seems to make sense also in non-euclidian geometries and can be applied to real life surfaces (using a string for compass; but I'm not sure how that might relate to some natural metric on a given geometry) I'm wondering how origami might be extended to non-euclidian geometry.

  29. I always heard that Euclid was able to trisect an arbitrary angle but he needed two marks on his straight edge. It would be nice to see a video on how he did that.

  30. I don't see, if you understand that last trick of origami, why one couldn't reconstruct those points of line intersections with euclidean symmetry. Isn't a fold just a line of symmetry?

  31. You can trisect any angle using a straight edge and a compass… given infinite time:
    let angle = 1;

    let trisect = 0;

    for (i=1;i<10;i++) {

    angle = angle / 2;

    if (i%2==0) {

    trisect = trisect – angle;

    }else {

    trisect = trisect + angle;




  32. The brides chair and euclids 47 problem can be solved with 3d folding.
    Proves 345 is the primer code for theory of everything.
    345 is the pixel template for reality of consciousness.

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