What is a Reuleaux polygon?

0:47 Welcome to Maths 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.

0:58 Hey Mira! Have you ever noticed that almost every single manhole cover on the street is round?

1:05 I have! You see them everywhere, right in the middle of the road or the pavement. Big, heavy iron discs covering the entrances to the sewers.

1:14 Which, by the way, are incredibly heavy and dangerous. So nobody listening should ever try to touch or lift one.

1:22 Definitely not. Leave that to the professionals. But just looking at them, it seems like a simple design choice. Why not square? Why not a triangle?

1:32 Yeah, a square would be much easier to draw and measure out. Buildings are square. Having stones are square. Why the sudden circle?

1:41 Because the circle is hiding a life-saving piece of maths. It is solving a massive, dangerous problem right under our shoes. A maths problem? In the sewer? Exactly! To understand why they are round, we actually have to imagine what would happen if we made a square manhole cover instead.

2:00 Okay, I am picturing a giant, heavy iron square. Right.

2:05 Now imagine a city worker needs to go down into the tunnels. They carefully lift the square cover off the square hole. Makes sense so far. But as they are moving it, the heavy square cover accidentally tips up on its edge, and it turns sideways, diagonally. Wait, does it fall in?

2:24 It completely falls in, crashing all the way down the tunnel. Hold on, that can't be right though. If the square hole is the exact same size as the square cover, how can it fit through?

2:37 Ah, close your eyes and picture a square. Think about the distance from one flat side to the opposite flat side.

2:44 Okay, got it.

2:46 Now think about the distance from one pointy corner all the way across the middle to the opposite corner.

2:53 The diagonal. Oh, the corners stretch further out. The diagonal of a square is longer than its flat side. Exactly!

3:01 The diagonal of a square is always longer than its side. So the hole's diagonal opening is long enough for the flat side to slip through.

3:10 Which means, if you tip the cover on its edge, its short flat side can slip right down into the wider diagonal part of the hole.

3:18 Whoosh, down it goes, which is incredibly dangerous.

3:22 So a round cover fixes that. How?

3:26 Think about what a circle actually is. A circle is the shape where every point on the edge is exactly the same distance from the middle.

3:34 Oh, so there is no long diagonal. There are no stretchy corners.

3:39 Exactly. It is the same width whichever way you measure it. Mathematicians have a real name for this.

3:46 They call it a curve of constant width. Constant width. So it's constantly the same distance across?

3:53 Yes. Turn around cover any way you like. Tip it on its edge, spin it around.

3:59 It never gets narrower than the hole. It just gets stuck. It can never, ever fall in. That is brilliant.

4:07 And there are huge practical reasons too. A circle doesn't have any corners, so when a worker slides it back on, they never have to twist it to line up the edges.

4:17 Oh yeah. Lining up a heavy iron square would be so annoying. You'd always get it slightly wrong.

4:24 Plus, because they are so unbelievably heavy, a round cover can just be tipped up and rolled away like a wheel. You can't roll a square.

4:33 Okay. I am never going to look at the street the same way again. The circle is the ultimate shape. It is amazing.

4:42 But here is the part that's going to bend your brain a little. Uh-oh. What? A circle is not the only shape that is the same width all the way round.

4:53 What? That's impossible. If it's the exact same width everywhere, it has to be a circle. That's what a circle is.

5:02 That is exactly what everyone thinks. Mathematicians like Leonard Euler studied them first, but in the 1870s, a German engineer named Franz Rolot showed the world something wild. Franz Reloud? What did he do? He studied how machines and gears move, and he looked closely at these strange, curved-sided shapes that are not circles, but they still have constant width.

5:27 How can it not be a circle? What does it look like? Imagine an equilateral triangle, three sides. But instead of the sides being straight, flat lines, imagine they bulge outwards, like the curves of a shield.

5:41 Pointy corners but bulging curved sides. Okay, I can see that.

5:47 If you draw the curves just right, no matter how you measure across it, it is the exact same distance. It is a curve of constant width.

5:57 Wait, so if I made a manhole cover in that weird, bulging triangle shape. It would never fall down the hole. It would get stuck perfectly, just like a circle. That is unbelievable. So it's pointy and wonky, but it acts like a circle. Exactly!

6:15 If you put a heavy book on top of rollers, shaped like Franz Roulot's bulging triangles, and push the book, it would glide totally flat. It wouldn't bounce up and down at all.

6:26 Because the rollers are always the exact same height, no matter how they spin. They have constant width. You've got it. We call them Ruleau polygons, after him. Okay, that is super cool. But where would I ever actually see one? I've never seen a bulging triangle wheel.

6:43 You might have seen one without realising it. Have you ever seen a British 50pence coin?

6:48 Or a 20pence coin? Oh, I have. They aren't round at all. They look like strange curved stop signs. Exactly.

6:57 They have seven sides. Seven outward curving sides. They are called rulo heptagons.

7:04 Heptigon meaning seven. But why make a coin that weird shape?

7:09 It makes them super easy to tell apart from all the circular coins, just by feeling them in your pocket.

7:15 Yeah, but if they aren't round, wouldn't they get stuck in vending machines or parking meters?

7:20 That is the magic! Vending machines measure the width of a coin to know how much money you put in. Because the 50 pence piece is a curve of constant width, no matter how it rolls in or spins, it always measures exactly the same width.

7:35 Oh wow! It rolls through the machine slot perfectly smoothly, even though it's covered in corners.

7:41 Yes, it tricks the machine into thinking it's completely round.

7:46 That is one of the coolest things I've ever heard. It's like a secret shape hiding right in plain sight. It really is.

7:53 Every time you see a round manhole cover or a strange multi-sided coin, you are looking at constant width.

8:00 Just a simple rule.

8:16 And the amazing thing is, mathematics fully understands and can classify every 2D curve of constant width.

8:23 Really? So mathematicians know exactly how to classify all those weird shapes that do the same thing? They do.

8:30 They have fully classified them. Sometimes an idea is so simple it turns out to be endlessly deep.

8:37 I am definitely checking my pockets for Rouleau Heptagons later. Alright Finn, time for the quiz. Are you ready? I am so ready. Bring on the shapes.

8:47 First question. Why are most manhole covers round instead of square? Next one. What do mathematicians call a shape that is the exact same distance across no matter how you measure it? Question three. Who was the German engineer from the 1870s who studied these special bulging shapes? Here's number four.

9:18 What happens to a square manhole cover if you tip it diagonally over its hole? Last question. Which British coin has seven curved sides but is still a curve of constant width? Great job! Let's see how you did. The first answer is so they don't fall down their own holes. For question two, it is called a curve of constant width.

9:49 The answer to number three is Franz Rouleau. For question four, it slips right through and falls in because the diagonal is longer than the side. And the final answer is the 50 pence coin or the 20 pence coin.

10:05 They are rouleau heptagons. I love that the street under our feet and the coins in our pockets are all secretly following this one beautiful maths rule.

10:15 It just goes to show maths isn't just on a blackboard. We walk on it and we carry it with us every day.

10:23 If you love learning about constant width today, please drop down and leave us a five-star review. Just scroll down and tap the stars.

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10:47 Thanks for exploring with us today. See you next time on Maths is not boring. Bye!