Well Euclid had the straight edge and the compass that those are the tools that he had available.
He didn't have verygood 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 rightanswer.
Let me just jump right in and show you how to trisect an angle byfolding paper.
What i'd like you to realise is that essentially youhave a straight edge because if you have two points that youpreviously constructed somehow then you can fold the paper so that thosetwo points are on the crease this is exactly the same thing asconnecting them by a straight edge.
Even better you can do perpendicularbisector in a single step because you can foldthe paper so that these two points cover each other and when you do that, that created crease will be perpendicular to the linesegment 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 aright angle with straight edge and compass which is probably an important pointto make.
It's not that there's no angle whatsoever that you can trisect it's that you can trisect an arbitrary angleso ninety degrees you can trisect because you can construct a thirty degree angle.
So we want to create anarbitrary 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.
Brady: you're gonna trisect that?Zsuzsanna Dancso: I am going to trisect that First we fold the paper in half so I justfold 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 thehalfway so now I have these two creases oneup here and one down here Now, here's the trick.
I'll mark two points one is this bottom of the angle Brady: YepZD: 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 createdbefore and this point halfway up the paper matches this line which is the angleitself 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 downand 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 iswhere this crease that I just created with this last fold where that intersects the bottom line so you might already see where this is goingBrady: 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 itmatches 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 yourcongruent 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 towhat it does to coordinates the same way that I told you about it inthe 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 wecan 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 equationwhich is the formula is is really really big and ugly and hard to plug into and you can solve it just by a simplefolding mechanism and then measuring the solution so I for example I showed you how to dosquare root two, I showed you how to do three I show you how to do one-third it turnsout that you can do all numbers that just involve fractionsand square roots and addition and subtraction but there'sa problem with cube roots so what this guy proved is that you will never be able to do cube roots.