The Surprising World of Polygon Tessellations: From Triangles to Pentagons

Have you ever noticed the repeating patterns on a tiled floor or the neat rows of honeycomb in a beehive and wondered why certain shapes fit together so perfectly? That magic trick is called a tessellation—covering a flat surface with shapes that leave no gaps or overlaps. In this post, we’ll explore why only three regular polygons—triangles, squares and hexagons—can do it, and how some sneaky five-sided shapes (pentagons!) have joined the party in unexpected ways.

Where did this come from?

The idea of tiling dates back to the ancient Greeks—Euclid even hinted at patterns in his Elements. Fast-forward to 1918, when German mathematician Karl Reinhardt proved only five families of convex pentagons could tile the plane. For decades, researchers chipped away at the puzzle until 2015, when a team finally tracked down the fifteenth and last type of convex pentagon that works. And guess what? That discovery popped up in real life on a street in Cairo long before anyone had the theory!

Where you'll see this in real life

1. Bee honeycombs: bees instinctively build hexagonal cells—hexagons pack together with no gaps and use the least wax for maximum storage. 2. Cairo tiling: stroll through Cairo’s old neighbourhoods and you’ll spot repeating pentagon patterns on pavements—an example of one of those 15 special families. 3. Computer graphics: 3D models often break surfaces into tiny triangles (called a mesh) because triangles are always flat—perfect for complex shapes. 4. Islamic art and mosaics: intricate star patterns and interlocking shapes rely on tessellations, mixing quadrilaterals, triangles and more to decorate walls and floors.

Why it matters at school

Studying which polygons tessellate helps you practise angle sums (remember, interior angles of an n-gon add to (n–2)×180°) and boosts spatial reasoning. It also shows the fun side of geometry—how maths sneaks into art, nature and technology. Next time you spot a pattern, you’ll know there’s a whole world of rules and history behind it!