Tile a Sheet Three Ways

Take a sheet of paper and try to tile it with a single regular shape. Equilateral triangles work. Squares work — that's ordinary grid paper. Hexagons work — the honeycomb pattern. Every other regular polygon fails: pentagons leave gaps, heptagons overlap.

Now compare efficiency. Draw a 1-inch edge hexagon, square, and triangle. The hexagon has perimeter 6 and area about 2.60. The square has perimeter 4 and area 1. The triangle has perimeter 3 and area about 0.43. Divide perimeter by area, and the hexagon wins cleanly — the least edge for the most space.