Tessellation for Kids: How Tiles Cover a Floor With No Gaps

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. Hey Mira, what are we talking about today? Finn, picture your bathroom floor, or a brick wall, or even the honeycomb inside a beehive. What do you notice about how they are built?

Well, they are made of shapes that fit together perfectly, like little squares in a bathroom, or rectangles on a brick wall. And the pattern just repeats over and over. Square, square, square.

Exactly! They cover a flat surface with no gaps, and their patterns repeat perfectly like wallpaper. But I want you to imagine a floor that goes on forever, stretching out past the horizon, and it is covered in just one single shape of tile, fitting together perfectly, with zero gaps anywhere.

Okay, I'm imagining it. A giant infinite tile floor made of one shape.

But here is the impossible part. No matter how far you walk across this endless floor, the pattern never, ever repeats. You can walk for a million miles, and the design will always look fresh and shifting.

Wait, what? Never repeats? How can one single shape do that? Wouldn't it have to start copying itself eventually?

That is exactly what mathematicians asked for decades. It's a mystery that involves a man cutting shapes out of cardboard on his kitchen table in Yorkshire, and a puzzle that lasted almost 50 years.

Okay, so let's back up to the bathroom floor. When tiles fit together with no gaps, that has a special mathematical name, right?

Yes, it's called a tessellation. You see tessellations everywhere in the world. Hexagons in a honeycomb are a tessellation, a checkerboard is a tessellation of squares.

I was looking at the paving stones on the sidewalk this week, and they were all triangles fitting together in a huge grid. It made a pattern that just copied itself forever.

Exactly! Squares, triangles, and hexagons are fantastic at tessellating. They create repeating, predictable patterns. But back in the 1960s and 70s, mathematicians started wondering: could you make a tessellation that covers the whole world but never repeats?

Why would it not repeat if it's just made of identical shapes? That feels impossible.

It does feel impossible. It's called an aperiodic tiling. Think of it like a jigsaw puzzle that builds outwards endlessly, but you can never pick up a massive chunk of it, slide it over, and have it match perfectly with another part. It's constantly shifting its design, never falling into a predictable rhythm.

That sounds dizzying. Did anyone figure out how to do it? In 1974, a mathematician in England named Roger Penrose found a brilliant way. He created a pattern that never repeated, but he had to use two different shapes working together. Two shapes? Like a puzzle with two different pieces. Yes, but that amazing discovery led to an even bigger question.

Mathematicians desperately wanted to know: could you do it with just one single shape?

One solitary shape that tessellates forever but never repeats its pattern.

Yes, they called this holy grail of geometry the Einstein problem. And Finn, it has absolutely nothing to do with the famous scientist Albert Einstein.

Then why call it the Einstein problem?

Because in German, the words Einstein literally translate to one stone or one tile. They were searching for that one magical stone.

So for decades, nobody could find this one stone. Did people start thinking it was just completely impossible?

A lot of people did. Finding a single shape that refuses to make a repeating pattern seemed almost like trying to break the rules of reality.

So who finally cracked it? Was it a famous professor sitting in a university with a giant supercomputer?

Nope. In November 2022, the answer was found by David Smith. He's a retired print technician living in Bridlington, a town in Yorkshire. He wasn't a professional mathematician. He just loved playing with shapes. Wait, a retired guy in Yorkshire? That's right. David Smith loved exploring geometry just for the pure fun of it.

He wasn't using complicated software at first. He was just printing shapes, cutting them out of paper and cardboard, and trying to fit them together on his table like a game.

That's exactly what I was doing this week. I was cutting out little paper robots to see if their arms would link together without any gaps.

It is exactly like that. Chance and discovery are often born from just playing. David Smith was playing with shapes, trying different angles, trusting his curiosity.

And he just stumbled upon the Einstein. The magical one stone.

He did. He found a shape with 13 sides. When you look at it, it has this jagged, uneven outline that looks a little bit like a shirt, or maybe a funny little hat. In fact, he lovingly called it the hat.

Haha, the hat. A 13-sided hat that never repeats. That is brilliant. But wait, how did he know it would never, ever repeat if the floor went on forever? He couldn't build an infinite floor in his living room.

That is the crucial question. He could see that it was doing something very strange on his table. It was locking together in clusters that stubbornly refused to settle into a simple repeating wallpaper pattern. But to prove for sure that it would never repeat out to infinity, David Smith needed some mathematical backup.

He reached out to a computer science professor named Craig Kaplan.

Ah, so this is where the professional mathematicians come in.

Yes! Craig Kaplan was absolutely amazed by what David sent him. Soon they were joined by two more mathematicians, Joseph Samuel Myers and Chaim Goodman Strauss.

So this whole team of four started intensely working on David's cardboard hat.

Exactly! And in March 2023, this incredible team officially announced their proof. They showed the world that the 13-sided hat can cover a flat floor forever, with zero gaps, and the pattern will truly never, ever repeat. The Einstein problem was finally solved.

That is completely wild. I'm just picturing a giant, infinite bathroom floor covered in these little hats, swirling around in unpredictable patterns forever.

There is one tiny detail though. To make the hat tessellation work without gaps, you occasionally have to flip the tile over. Like taking a right-handed glove and turning it inside out to make a left-handed glove, creating a mirror image.

So you need the hat, and sometimes it's mirror image.

Exactly, but the story doesn't end there. That same team kept working, and later in 2023, they found another shape they called the spectre. And the spectre can tile forever, without repeating, and without ever needing to be flipped over.

Wow, they really did it. They completely conquered the one-stone problem from every angle.

They really did. But what I love most is that this massive mathematical breakthrough didn't come from a dusty textbook. It came from a man simply playing with shapes on his table, trusting his curiosity and looking closely at the world.

It really makes you look at the world differently, doesn't it? I keep thinking about how many shapes are just sitting out there, waiting for someone to cut them out and notice what they can do.

That is the true beauty of it. Maths isn't just a list of numbers you memorise in a classroom. It's the secret rules hiding in bathroom floors, honeycombs, and brick walls.

Now I'm going to be staring at every tiled floor I walk across, checking to see if it repeats. I'll be the kid crawling on the floor at the grocery store.

You absolutely should! The fact that an infinite, non-repeating pattern can exist from just one simple shape blows my mind. It's like the universe has a hidden setting for beautiful, endless chaos.

Okay, do we have any fun facts about the hat tile before we get to the quiz?

Yes. Did you know the 13 sides of the hat are actually made by connecting eight smaller shapes called kites? That's why it has such a strange, jagged outline instead of smooth, straight edges.

Eight kites glued together into one big tile. That is so specific.

It is! And the discovery of the hat was considered one of the biggest mathematical breakthroughs of the entire decade, making David Smith a total legend in the maths world. Not bad for a retired print technician playing with cardboard.

I love that so much. He just followed the fun of the puzzle, and it changed math history forever. Alright, Finn, it is time for the quiz. Are you ready? I am so ready. Let's do this.

First question. What is the mathematical word for shapes fitting perfectly together to cover a surface with no gaps? Next one. What does the German word Einstein literally mean when mathematicians used it for this puzzle? Question three. Who was the retired print technician from Yorkshire who found the magical shape in 2022?

Here's number four. What did he name his 13-sided shape because of how it looked? Last question. Who found a never-repeating pattern using two shapes back in 1974? Great job! Let's see how you did. The first answer is tessellation. For question two, it means one stone or one tile. The answer to number three is David Smith.

Question four, he called it the hat. And the final answer is Roger Penrose.

You know, I don't think I'll ever look at a jigsaw puzzle or a tiled floor the same way again. The idea that one simple shape could go on forever without ever repeating. It's just brilliant.

It really is. It reminds us that there are still incredible mysteries waiting to be found in the simplest things around us.

Hey, if you enjoyed exploring the hat tile with us today, please leave us a five-star review. Just scroll down on your app and tap the stars. It really helps other people find the show.

Yes, and make sure you tap follow on this podcast so new episodes turn up on their own. Oh, and we have sister shows too. History, science, and philosophy is not boring. Go give those a listen and a follow as well.

And if you want a shout out on the show or just want to say hi, email us at hello at kidopoly.com.

See you next time on Maths is not boring. Bye.

Tessellation for Kids: How Tiles Cover a Floor With No Gaps

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