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Geometric packing 2026-10-06

Placement of geometric objects without interior overlap. A packing fraction measures how much area or volume they occupy. Circle packing and sphere packing are examples.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 74 / 3 / a / Solution Created 2026-10-03 Updated 2026-10-06

A square circle packing of equal radii has one circle per cell of side

2 r_{1}

. Its packing fraction is therefore

η_{square} = \frac{π r _{1}^{2}}{( 2 r _{1} ) ^{2}} = \frac{π}{4} .

For the alternating-radius square circle packing shown in the source figure, let

d

be the nearest unlike-centre separation. A two-circle repeat cell has area

2 d^{2}

π (r_{1}^{2} + r_{2}^{2})

. Unlike circles require

d \geq r_{1} + r_{2}

; like circles on the two sublattices require

2 d \geq 2 r_{1}

and

2 d \geq 2 r_{2}

. Thus

η = \frac{π ( r _{1}^{2} + r _{2}^{2} )}{2 max {( r _{1} + r _{2} ) ^{2} , 2 r _{1}^{2} , 2 r _{2}^{2} }} .

All three nonoverlap constraints matter when varying the radius ratio.

Take

r_{1} \geq r_{2}

as in the illustrated arrangement and set

s = r_{2} / r_{1}

. If

0 \leq s \leq 2 - 1

, the large circles determine the spacing and

η = (π /4) (1 + s^{2})

increases with

s

. If

2 - 1 \leq s \leq 1

, unlike contacts determine it and

η = (π /2) (1 + s^{2}) / (1 + s)^{2}

, whose derivative is proportional to

s - 1

and is nonpositive. The maximum is therefore at the joining point:

\frac{r _{1}}{r _{2}} = 1 + 2, η_{m a x} = π (1 - \frac{1}{2}) ≃ 0.92015.

The large circles then touch their large neighbours, while each small circle exactly fills a square interstice. Interchanging the species gives the reciprocal radius ratio. This is the highest packing fraction within the specified alternating square arrangement; it is not a claim about every possible unequal-circle packing.

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