You have a 4 m × 4 m garden bed. Tomato plants need 50 cm of space. How many can you fit in rows? In a hexagonal pack?
Square pack vs hex pack
A square grid wastes a little space — each plant sits in a 0.5 × 0.5 box. Offset every other row by half a step and the rows can squeeze 0.43 m apart instead of 0.5 m. Same plants per row, but more rows fit: about 12% more tomatoes from the same bed.
Bees building honeycomb and oranges stacked at the market both use the hexagonal pack for the same reason — it's the densest way to fit equal circles into a space.
A 3 m × 6 m bed, plants 0.5 m apart in a square grid. How many plants?
Square grid
Each plant takes a 0.5 × 0.5 cell. 4 m / 0.5 = 8 per side. 8 × 8 = 64 plants.
Hexagonal pack
Rows offset by 0.25 m. Each row still has 8 plants but rows are 0.43 m apart. ~9 rows. About 72 plants — 12% more produce.
Bees and oranges pack hexagonally for the same reason: it's the densest possible arrangement of equal circles in a plane.