Circle packing studies how circles can be arranged in a plane so they cover as much area as possible without overlapping. It matters in geometry because it connects simple shapes to optimization, symmetry, and proof. The same ideas also appear in nature and design, from bubbles and seeds to coins, pipes, and cellular materials.
Comparing square and hexagonal packing shows how small changes in arrangement can greatly affect the empty space left between circles.
In square packing, each circle touches four neighbors and the centers form a grid of squares. In hexagonal packing, each circle touches six neighbors and the centers form a pattern of equilateral triangles. Hexagonal packing is the densest possible packing of equal circles in the plane, with density pi divided by the square root of 12, about 0.907.
This means about 90.7 percent of the plane is covered by circles, while the remaining space is made of small curved gaps.
Key Facts
- Packing density = area covered by circles / total area of the repeating cell.
- Square packing density = pi/4 ≈ 0.785, so about 78.5% of the plane is covered.
- Hexagonal packing density = pi/(2 sqrt(3)) = pi/sqrt(12) ≈ 0.907, so about 90.7% of the plane is covered.
- In square packing, each circle touches 4 neighboring circles.
- In hexagonal packing, each circle touches 6 neighboring circles.
- For circles of radius r, circle area = pi r^2 and diameter = 2r.
Vocabulary
- Circle packing
- Circle packing is the arrangement of circles in a region or plane so that they do not overlap.
- Packing density
- Packing density is the fraction of the total area covered by the circles.
- Square packing
- Square packing is an arrangement where circle centers form a square grid and each circle touches four neighbors.
- Hexagonal packing
- Hexagonal packing is an arrangement where circle centers form triangular rows and each circle touches six neighbors.
- Unit cell
- A unit cell is a small repeating region that can be used to calculate the density of a repeating pattern.
Common Mistakes to Avoid
- Using the circle area alone as the packing density is wrong because density compares circle area to the area of a repeating cell or region.
- Assuming square packing is densest is wrong because shifting alternate rows lets circles fit into gaps and creates the denser hexagonal packing.
- Counting every circle in a unit cell as a whole circle is wrong because circles on edges or corners may be shared with neighboring cells.
- Comparing densities using different circle radii is wrong because packing density depends on arrangement, not the absolute size of equal circles.
Practice Questions
- 1 A square packing uses circles of radius 2 cm. One repeating square cell has side length 4 cm and contains one circle. Find the packing density as a decimal.
- 2 A hexagonal packing of equal circles has density pi/(2 sqrt(3)). Approximate this density using pi = 3.14 and sqrt(3) = 1.732, then state the percent of area covered.
- 3 Explain why hexagonal packing leaves less empty space than square packing even though the circles have the same size.