Platonic Solids for Kids: Why There Are Only Five Perfect Dice

0:47 Welcome to Maths Is Not Boring by kidoply.com. I'm Mira. And I'm Finn. Before we start, a huge hello to Desmond and Rowan in Port Credit, Ontario, Canada, who listen every night before bed. Thanks for exploring with us. Yeah, thank you, Desmond and Rowan. If you haven't already, tap follow wherever you're listening so you never miss an episode.

1:11 You know, since Desmond and Rowan listen right before bed, maybe they play a board game first. Which usually means rolling dice. Exactly.

1:21 But have you ever looked really closely at a die, Finn?

1:25 Well, yeah, it's just a cube. Six sides with dots on them. You roll it to see how many spaces you get to move.

1:32 But what if I told you there are exactly five perfectly regular shapes in the entire universe, and each one makes a beautifully fair die. Only five. And no matter how clever you are, nobody will ever find a sixth.

1:47 Wait, what? Only five perfectly regular shapes in the whole universe. That can't be right.

1:53 It is absolutely true. A perfect die has to be completely fair. Every single face has to be exactly the same regular shape. Like a perfect square or a perfect triangle. Not stretched or squished at all.

2:09 Right, so whichever way it lands on the table, it has the exact same chance of turning up.

2:15 And the exact same number of faces must meet at every corner, popping outwards without caving in or crossing themselves. If you do that, you get a mathematically perfect shape. Today we call them the platonic solids.

2:31 Platonic solids? That sounds ancient.

2:34 It is! Around the year 380 BCE, in the city of Athens, a mathematician named Theotetus figured out how to build and study them.

2:45unknown Haha!

2:46 Theotetus! That is a brilliant name. Was he just sitting around making dice for games?

2:52 He was exploring geometry. He did the really hard work of proving how they work. A little later, around 360 BCE, his famous friend Plato wrote about these shapes.

3:05 Oh, and that's why they got named after Plato instead. Exactly.

3:09 People loved Plato's writing, so the name stuck. But Theotetus is the one who really saw them clearly first.

3:17 Well, shout out to Theotetus! So what actually are these five shapes?

3:22 Let's build them in our minds. Start with the simplest flat shape you can think of: a perfect triangle, where all three sides are the exact same length.

3:33 Okay, I've got a flat triangle in my head.

3:36 Now, to make a solid 3D shape, we need to fold them up so they meet at a corner. If you take three triangles and fold them together at a single point, what does it look like? Like the top of a pyramid. Exactly. Add a flat triangle to the bottom and you have a four-sided pyramid. That is the first perfect shape.

3:59 It has four faces and exactly three triangles meet at every corner.

4:04 Wow, okay, a four-sided die. I've actually seen those in games. What if we use four triangles at every corner instead of three?

4:14 If you stick four triangles around a corner, you get a shape that looks like two pyramids, stuck bottom to bottom. It has eight sides. Oh, an eight-sided die. I know those too. That's the octahedron. Now try five triangles at every single corner. This one balloons out into a beautiful roundish shape with 20 triangle faces.

4:40 Twenty sides? A 20-sided die? That's the one you roll for a really big move. So that's three shapes down. What if we put six triangles together at a corner?

4:52 Here is the mind-bending part. Try to picture six perfect triangles meeting at one single point.

5:02 Okay. Wait. If I put six of those triangles in a circle, they just lie completely flat. Like a pizza sliced into six pieces.

5:12 Exactly! They lie perfectly flat. There is no gap left to fold them up into a 3D shape. Oh!

5:21 So you literally cannot make a 3D shape with six triangles at a corner. That's it for triangles.

5:28 That's it! So let's try perfect squares. Three squares at a corner folds up perfectly into a cube. Our regular six-sided die.

5:38 And four squares? Wait, four squares would lie completely flat like a checkerboard. You can't fold that either.

5:46 You've got it! So there is only one perfect shape made of squares. Now try pentagons, which are five-sided shapes. Put three of those at a corner and it folds into a beautiful twelve-sided shape. A twelve-sided die? Yes, the dodecahedron. And if you try hexagons, which have six sides, just three of them lie perfectly flat on the table.

6:11 So there is no room to fold them. You literally just run out of space to make any more shapes.

6:17 Exactly. And around 300 BCE, a great mathematician named Euclid wrote a famous book called The Elements in the City of Alexandria. In book 13, he ended the whole thing by showing exactly what we just saw. There can never, ever be a sixth perfect solid that pops outwards.

6:39 That is so cool. You just literally run out of room to fold them. The corners just lie flat.

6:46 Exactly. Once you see that, you realise these five shapes are incredibly special. Plato actually thought these five shapes were the hidden building blocks of the whole universe. Wait, really? Like everything was made of them? Yes. He imagined the four-sided pyramid was fire, because its corners are sharp and pointy.

7:10 The cube was earth because it's completely solid and stacks so well. The eight-sided octahedron was air. The twenty-sided icosahedron was water, because it rolls almost like a ball. And the twelve-sided dodecahedron was the shape of the heavens. That is such a beautiful idea. Is it true? No, it's not real science.

7:36 It was just Plato's idea. But it shows how deeply people felt the magic of these shapes when they first saw them.

7:43 I love that. Do these perfect shapes actually show up in nature though?

7:48 They really do. If you look at ordinary salt crystals under a microscope, they naturally form into perfect little cubes. No way! Salt is tiny cubes. Yes, and even wilder, the shells of some viruses are shaped exactly like 20-sided icosahedrons, because it's the absolute perfect way to build a strong, fair, protective bubble.

8:13 Alright Finn, quiz time! Okay, I'm ready. First question. What is the name of the mathematician from Athens who first studied these perfect shapes? Next one. Which famous thinker were the solids actually named after? Question three. If you put three squares together at every corner, what shape do you make?

8:46 Here's number four. What happens if you try to put six triangles together at a corner? Last question. How many of these perfect shapes did Euclid prove there are in the whole universe? Great job! Let's see how you did. The first answer is the mathematician was Theotetus. For question two, they were named after his friend, Plato.

9:18 The answer to question three, three squares make a six-sided cube. For question four, six triangles will just lie perfectly flat so they can't fold into a 3D shape. And the last answer is, there are exactly five.

9:35 I am never going to look at a normal board game die the same way again.

9:40 I know, right? Every time you roll a die, you're holding one of the only five perfect shapes in the universe.

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10:18 See you next time on Maths Is Not Boring. Bye, everyone.