Tessellation — introduction and transformations

Have you ever looked at a tiled floor, a honeycomb, or a brick wall and noticed how the shapes fit together perfectly with no gaps and no overlaps? That's tessellation in action!

Tessellations are everywhere in the world around us — from the hexagonal cells in a beehive to the square tiles on a bathroom floor to the decorative tilework in mosques and palaces. Mathematicians study tessellations because they reveal deep truths about how shapes behave and how space can be filled.

The word tessellation comes from the Latin word "tessella", meaning a small square tile — the kind used in ancient Roman mosaics.

A regular tessellation uses only one type of regular polygon (a shape where all sides and all angles are equal) repeated over and over.

Here's the surprising fact: only three regular polygons can tessellate on their own:

• Equilateral triangles (interior angle = $60°$)
• Squares (interior angle = $90°$)
• Regular hexagons (interior angle = $120°$)

Why only these three? For shapes to tessellate, the interior angles meeting at each vertex (corner point) must add up to exactly $360°$.

• Equilateral triangles: $6\times 60°=360°$ ✓
• Squares: $4\times 90°=360°$ ✓
• Regular hexagons: $3\times 120°=360°$ ✓