Fold and Cut Theorem - Numberphile

I have a pair of scissors 'cause I want to talk about a piece of maths called the Fold and Cut Theorem.

It's one of those things in maths where they just name it what it is.

This first came to my attention when I was trying to cut a square hole

in a piece of paper. So, like, my first instinct - as a person who cuts holes in paper - was to jab a hole and then cut round the edge of the square.

But I thought, ok maybe there's an easier way to do this,

and kind of my mathematical brain kicked in,

and I realised that if I fold this in half like that,

I can just cut along those three edges.

Ok, I've roughly approximated the hole I was originally trying to draw.

There's a square hole, and I guess

like, the mathematician in me was like, "Can I do that more easily? Can I make this more efficient?"

So what I thought then was, if I fold this twice,

so I make the first fold the same,

and then fold that at 90 degrees to that,

I now have this bit of line,

and I can now cut two cuts like that and,

I'll end up with a square hole!

Yeah, it's two cuts it's less than three cuts

it's more efficient

but of course you want to optimise this

so your mathematically efficient way of doing this with just a square

is to start off with one fold in half,

a second fold like that,

and then your third fold,

is a diagonal fold through that corner.

So as long as you get all of those kind of folds exactly right,

and the lines that you're cutting along are all lying on top of each other,

you can now cut out the entire square, the whole shape,

with just one straight cut,

like that.

And this is the moment of truth, the bit where you don't really believe what you've done has worked,

but it turns out a square is pretty much the easiest one to do!

It is efficient and it's one straight cut

And this kind of idea like I went to the pub as you do when you've discovered a nice thing,

and I was like I wonder if this is a thing you can do in general, I wonder what shapes you can cut out with one cut

and a friend of mine who was in the pub said isn't that a theorem?

As all mathematicians will tend to do I went and looked up that

It is a theorem, it's called the Fold and Cut Theorem and it says that any shape whose edges are all made up of straight lines,

can be cut out with one cut, if you're prepared to fold a bit of paper.

which is just insane!

and I love this it was proved by a guy called Erik Demain and a team of other researchers,

they've proved that it is possible to cut out any straight line edge shape,

even if it's got more than one bit to it, even if it's got a hole in the middle,

you can make one cut and cut out the whole thing!

There are various examples of this throughout history

So Harry Houdini, the magician, used to apparently do

a star with one cut as part of his magic show

and it is sufficiently impressive to go in a Houdini magic show, it turns out

So I've looked that up and I've figured out how to do that

So it's in half first,

It's... in half that way and in half that way

To get a cross shape

And then the fold is there

And in theory that should give you 108 degrees-ish

You can then fold that one in half

Fold that back and fold that

And you can tell you've done it right 'cause that's the same size as that, ish

And there's actually a nice little story behind this

because apparently when America was founded in the 1770s

they needed a flag and they went to see a seamstress called Betsy Ross

I don't know if it's a true story, could be one of these made-up stories

that George Washington walks into a fabric shop and was like "Gonna need a lot of stars cutting out"

and she had a piece of fabric and she just grabbed it and folded it up like this

and she did the one cut, so it's from there, kind of up that way

and she unfolded this and showed George Washington

that she cut out a perfect five-pointed star with just one cut

and this, this story goes that he was like "oh well, you can make all the flags then."

And she then had like work for life making all of the American flags

and at that stage there were only like a few stars on the flag

they kept updating it and adding more stars so she had to remake all the flags

so it was quite a decent amount of work just 'cause she happened to know this little trick

and I love that, I loved it

The most, like the most impressive thing for me

is that not only have you got a star, you've also got a piece of paper with a star-shaped hole in it

That's a five-pointed star. That's my favorite bit, is that you get the opposite thing as well

'cause it is just that line that's cut, it's nothing else

The next thing in the paper that they list in, kind of, things that they've seen

was that someone had once seen an example of someone who could cut out any letter of the alphabet on demand

and I looked this up and tried to find it and I couldn't find anyone who did this

so I was like "right, well that's... I'm gonna do that,

I'm gonna figure out how to do every letter of the alphabet."

And it turns out it's not even just a cool thing to be able to do

it's actually a really fun thing to work out

So you start with the simple letters like L

'cause they're quite easy, that's just one, kind of, diagonal fold

and then you need to make the ends

You can do things like O as long as you don't mind it being quite square.

And then things gradually get more and more difficult

and the easier ones tend to be quite symmetrical.

So T is quite easy.

S is not easy! S is a bit of a nightmare, but it is possible.

So I've worked out a fold pattern for every letter of the alphabet.

Having done this, I then got an email from someone saying, "I've just found this pdf

with a fold pattern for every letter of the alphabet."

I was like, what!?

That already existed. Oh, well, never mind.

But, you know, the system that I've worked out I'm happy with.

It's slightly different in places.

But you know, there are these things out there,

but it is more fun to work it out yourself.

Well, we've got A B C D E F G H I J and K

And then, L M N O P Q R S T U V W X Y and Z

That's an A

B

That's C

Here's D

E

F

G ... that's the closest I can get to G without any curves

H

There's I

J

my initial ... K

That is completely different from J ... that's L

That one is M

there's no way I'll get this the right way round ... N

O

P ... it's good enough

it's tiny, but it's a Q

That's R

Brady: "nice!"

A bit of a nightmare that one.

Brady: "well done!"

S

Slightly funky T

U

Slightly '70s looking V

W

X ... it's quite a fun one to work out

Very happy ... that's a Y

It's looking good from there, isn't it? So that's Z

Like, working out how to fold that is such a mathematical thing to do.

Like, it's the type of thinking that mathematicians use can be applied to so many situations,

and this is exactly one of them.

But this is an example of something which also mathematicians have properly used, you know,

mathematical analysis on, and they've proved a thing about it,

which is, you know, for me, that is what maths is about.

It's not about calculating things, it's about proving things.

And it's about understanding how the universe works.

And it turns out that in this universe, one cut suffices.

We'd like to thank lynda.com for supporting this video.

Now Lynda's great. They have a huge range of video courses.

I was actually going through the catalog earlier, and came across some great ones about how to use typefaces and typography.

This is something I find really interesting.

This one here is called The 33 Laws of Typography, and it's gone right to the top of my watch list.

I'll be having a look at that one soon, I think.

But whatever it is you want to learn, whether it's something creative, techy, maybe just getting better at business, Lynda will have something for you.

There's over three thousand video courses, and one hundred thousand tutorials in the vault.

You can sign up now for a free ten-day trial. That's unlimited access to all those videos by going to lynda.com/numberphile

or clicking on the link in the description.

That's Lynda, L-Y-N-D-A dot com slash numberphile

Give it a try. And our thanks to Lynda for supporting numberphile.

I did a show at Cheltenham a couple years ago, and

one of the things that I did in it was, as the challenge, like, to pick a person in the audience, ask them what their name was,

and then cut out that word while explaining the maths behind this.

It was brilliant. I asked for a four-letter word. I said, has anyone got a four-letter name?

And this guy shouted his name, and I can't remember what it was, but it was five letters.

And everyone was like, mate, that's five letters. And it was just a gift, it was hilarious.