What is the math of origami?

0:47 Welcome to Mass is not boring by kidopoly.com. I'm Mira! And I'm Finn! If you haven't already, tap follow wherever you're listening so you never miss an episode.

0:59 Wait, we're talking about paper today. Like ordinary everyday paper.

1:05 We are. Imagine you fold a paper crane. You take a crisp square of paper, you crease it, you bend it, you fold the little wings down, and you press it totally flat onto the table.

1:19 Yeah, I've made those. You get little paper cuts if you're not careful, and sometimes the beak comes out looking a bit squished.

1:27 But imagine you took that perfectly flat crane and you unfolded the entire thing. If you flatten it out completely back into a square and then open it back up, the paper is covered in an intricate map of creases.

1:41 Yeah, it looks like a bunch of zigzags all over the place.

1:45 But hiding inside those criss-crossing lines is a secret rule. A math rule just sitting right there in the piece of paper.

1:53 A rule? Like the paper has to obey it.

1:57 Exactly. A rule so strict that every single flat origami shape in the world has to follow it. And a trick so clever it solved a 2,000-year-old impossible mystery.

2:09 Okay, so what are we actually looking at when we unfold the crane?

2:13 If you flatten your paper out on a table, you'll see the crease lines meeting at little corners, which mathematicians call vertices. And if you look closely, there are two types of creases.

2:26 Right, because some fold upwards toward you and some fold downwards away from you.

2:31 Exactly. The ones that stick up toward you are called mountains. The ones that dip down away from you are called valleys.

2:39 Mountains and valleys. That makes total sense. You can literally feel the ridges and the ditches if you run your finger lightly over the paper.

2:47 For centuries, people just folded paper by feel and tradition. They made beautiful shapes, but they didn't think of it as mathematics. But in the 1980s, a Japanese physicist named Jun Mayakawa started looking really closely at those mountains and valleys. What was he actually puzzling over? He was designing complicated, incredible origami models, and he wanted to know if there was an underlying invisible structure to how the paper behaves.

3:17 Around the same time, a French mathematician named Jacques Justin was wondering the exact same thing independently. Did they find something hidden in the creases? They noticed a pattern, not something they invented, but a mathematical truth that had been quietly hiding in the paper all along. They saw that wherever creases meet at a point in a model that folds completely flat, the mountains and valleys always follow a very strict rule.

3:46 Wait, a strict rule? What does it actually say? I'm picturing all these zigzaggy lines meeting at one spot.

3:54 This rule is now called Maikawa's theorem. It says that if you count the number of mountain creases meeting at a point and count the number of valley creases, those two numbers will always differ by exactly two. Wait, what?

4:11 Always exactly two, no matter how many creases there are.

4:16 Always exactly two. If you have five mountain creases meeting at a point, you could have exactly three valley creases, or seven. Or if you have three valleys, you could have one mountain or five. The difference is always two.

4:33 That just can't be right though. What if I take a piece of paper and just scrunch it or fold it a hundred times in crazy random directions?

4:43 Ah, but remember the golden rule. It only applies to models that fold flat. If it folds perfectly flat, without crumpling, and you count the creases meeting at any one vertex, the difference between the mountains and the valleys will definitively be two.

4:59 It's just locked in. Like a secret code written into the very fabric of the paper.

5:06 It is totally locked in. It's a fundamental law of geometry that the paper simply cannot break. And that's not the only rule hiding in a flat fold. There's another famous one called Kawasaki's theorem. Oh, there's more. What does that one do? If you measure the angles between the creases around that single point, all the alternating angles add up to exactly a straight angle, which is 180 degrees.

5:34 So the paper isn't just bending randomly. It's actually doing serious geometry every single time you squash it flat upon the table.

5:42 It is. And because paper is secretly doing geometry, it can do things that were supposed to be completely impossible. Impossible how? Let's travel back to France in the year 1837. A mathematician named Pierre Vansel was looking at a puzzle that people had been trying to solve for 2,000 years.

6:05 2,000 years is a really long time to be stuck on a single puzzle. What was it?

6:11 It was a problem about splitting an angle. If you draw an angle on a page, it is very easy to cut it exactly in half using just an unmarked straight ruler and a drawing compass.

6:24 Right. You just use the compass to find the middle and draw a straight line down the centre. I've done that in school.

6:31 But for two millennia, people wanted to split any arbitrary angle into three perfectly equal parts. This is called trisection. The greatest minds tried everything they could to do it with just a straight edge and a compass. Did Wonsell finally figure out the secret way to do it? He figured out that it absolutely can't be done.

6:55 In 1837, Pierre Woncel proved mathematically that it is permanently impossible. The tools just don't allow it.

7:05 Oh, so everyone was staying up late trying to do something that literally could not be done in the universe.

7:12 Exactly. But here is the truly wild part. Von Sel proved you couldn't do it with a ruler and compass. But you can do it with a simple sheet of paper.

7:24 Wait, a piece of paper can solve a math problem that everyone thought was impossible.

7:30 Yes. In 1980, an origami master named Hisashi Abe showed that with a few clever specific folds, a plain sheet of paper can perfectly trisect any acute angle. The paper can line things up in a way a ruler simply cannot.

7:49 Haha, take that ruler and compass. Paper just sneaks past the rules and solves it.

7:55 The paper doesn't just hold a pretty shape fin, it literally holds the answer. When you fold it, the paper is secretly calculating things for you.

8:05 That is so incredibly cool. I'm never going to look at a folded note or a paper aeroplane the same way again. They're all little math machines.

8:14 And these hidden folding rules aren't just for fun party tricks or paper cranes. They are so powerful they actually go to space. Space? Like on rockets and satellites? Yes. In 1970, a Japanese astrophysicist named Koryo Miura invented a special folding pattern called the Miura fold. It's designed so that a giant flat sheet can open out completely or close up tightly with just one single pull on its corners.

8:46 Oh, that's like those Taurus maps you can buy that just snap open and shut in one second without getting all tangled up.

8:53 Exactly like those, but on a much bigger scale. In 1995, the Miura Fold flew on Japan's Space Flyer unit. They used these exact folding rules to pack giant solar panels tightly into a rocket and then unfurl them perfectly once they were out in orbit.

9:14 So the exact same math that makes a paper crane sit flat on my desk helps power a spaceship in orbit? That is amazing. Alright Finn, it's quiz time! Okay, I'm ready. I've got my flat-folded brain prepared. First question.

9:30 What do we call the creases that fold upwards toward you? Next one. What Japanese physicist noticed the strict rule about mountains and valleys in the 1980s? Question three. In a flat foldable model, the number of mountain and valley creases meeting at a point always differ by exactly what number? Here's number four.

10:03 What French mathematician proved in 1837 that you can't trisect an angle with just a ruler and compass. Last question. What is the name of the special spacefold that flew on Japan's spaceflyer unit in 1995? Great job! Let's see how you did. The first answer is, the creases that fold toward you are called mountains.

10:36 For question two, the physicist who noticed the pattern was Jun Maikawa. The answer to question three is they always differ by exactly two. For question four, it was Pierre Wontzel who proved it was impossible. And the final answer is, that special folding pattern is the Miura fold.

10:58 I honestly still can't get over the fact that mountains and valleys always differ by two. It's like the paper just secretly knows.

11:07 I love that too. It makes something as ordinary as folding a piece of paper feel like you're holding a secret code.

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11:45 See you next time on Maths Is Not Boring.