What are hyperbolic tessellations?

A hyperbolic tessellation is like a puzzle made from shapes that curve and stretch on a special kind of paper, not flat, but wobbly, like a jellyfish.

Imagine you have a toy with tiles, squares, triangles, or hexagons, that fit together perfectly on your floor. That's a tessellation. Now picture the floor isn't flat anymore; it’s curved in a weird way, like if you stretched out a balloon and painted shapes on it. This is called hyperbolic geometry, and when those tiles fit together on this strange surface, that’s a hyperbolic tessellation.

Think of a sponge, not the kitchen kind, but one that has tiny holes everywhere. If you put shapes like triangles or hexagons in those holes and make them repeat, it's like a hyperbolic tessellation. Every corner connects to many more corners than on flat paper.

On flat ground, a triangle only has 3 sides, and they fit just right with other tiles. But on this wobbly floor, each triangle can have more space around it, like if every kid in the playground had 6 friends instead of 3 or 4. That’s how shapes behave in hyperbolic tessellations: more of them fit together in a pattern that never ends, just like an endless game of tag!