Some shapes can tile a flat surface without gaps — hexagons, squares, and triangles all do this. What sets hexagons apart is efficiency: using algebra and geometry, you can prove that hexagons use less boundary material to enclose the same area than squares or triangles do. Bees exploit exactly this property, building hexagonal honeycombs rather than square or triangular ones to minimize the wax they need.

How much material a shape needs depends directly on the flat space it has to cover. In the Tiling Patterns project, you prove that hexagons use less wax than squares or triangles to create the same area. That result explains why bees choose hexagons — they get the most storage space from the least building material.

Bees build hexagonal honeycombs because hexagons tile a flat surface with no gaps or overlaps — using less wax than squares or triangles to create the same area. This project tests that principle hands-on. You build pentominoes from LEGO bricks and heptiamonds from pattern blocks, then cut wheelbarrow and kite-and-dart pairs from tagboard to experiment with tiling rectangular grids and flat planes. Twelve pentominoes can fill several rectangular grids, and all but one of the 24 heptiamonds can tile a plane on their own. Using algebra and geometry, you then prove why the hexagon's efficiency holds up mathematically.

Why do bees build hexagonal honeycombs instead of squares or triangles? This project explores exactly that question by testing which shapes can tile a flat surface without gaps. Using LEGO bricks, pattern blocks, and tagboard, you build pentominoes, heptiamonds, wheelbarrow pairs, and kite-and-dart pairs, then experiment with tiling rectangular grids and planes. Twelve pentominoes tile several rectangular grids, and all but one of the 24 heptiamonds tile a plane on their own. With algebra and geometry, you then prove that hexagons use less wax than squares or triangles to enclose the same area — which is precisely why bees rely on them.