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
