Regular hyperbolic tiling

Do you like puzzles? Regular hyperbolic tiling is like a puzzle that has many shapes, called tiles, that fit together perfectly to cover a whole floor or wall. But this puzzle is different from the ones you may have seen before because it is not flat like your table or the floor. It is a puzzle that is made for a curved space!

Now, imagine you are in a giant ball, and this ball is made of a special material. This material is like a special mirror that can reflect everything that gets inside it. That means, if you look around, you will see yourself many times, like in a house with many mirrors.

Regular hyperbolic tiling is like putting many shapes inside this ball, like triangles, squares, or hexagons, and making them all fit together perfectly. But there is a trick here! Remember that the ball is curved, so these shapes need to be curved too, like a slice of pizza.

Another thing about this puzzle is that, the more tiles you add, the more space they will cover, and the smaller they will become. In fact, there is no limit to how many tiles you can add, and they will never cover the whole ball completely. This means that some parts of the ball will always be empty, no matter how many tiles you add.

Regular hyperbolic tiling is not just a fun game, it's also very useful in math! It helps us understand shapes that exist in a curved space, like the surface of a ball or a donut. It's like exploring a new world, where the rules are different, and the puzzles are more challenging, but also more fascinating!