Packing Intuition Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

Compare the packing efficiency of square packing (

\pi/4

) vs hexagonal close packing (

\pi/(2\sqrt{3})

) of unit circles. Which is more efficient and by how much?

1
Step 1: Square packing efficiency: $\dfrac{\pi}{4} \approx 0.7854$ (78.54\%).
2
Step 2: Hexagonal packing efficiency: $\dfrac{\pi}{2\sqrt{3}} = \dfrac{\pi\sqrt{3}}{6} \approx \dfrac{3.1416 \times 1.7321}{6} \approx 0.9069$ (90.69\%).
3
Step 3: Difference: $90.69\% - 78.54\% = 12.15$ percentage points. Hexagonal is more efficient.

Answer

Hexagonal:

\approx 90.7\%

; Square:

\approx 78.5\%

. Hexagonal is

\approx 12

percentage points more efficient.

Hexagonal close packing is the densest packing of equal circles in the plane (proven by Thue's theorem). Each circle is surrounded by

6

neighbours instead of

4

in square packing, leaving smaller gaps and achieving higher density.

About Packing Intuition

Arranging objects of given shapes to fit as many as possible into a bounded region without any overlapping.

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