packing in the toilet rolls

The TV show QI is a marvellous source of trivia.  A while back I heard them say that Sainsbury's is shrinking the size of the tube through the centre of toilet rolls to save volume, and hence the number of delivery lorry trips needed.  A quick Google confirms that this is the case: the diameter of the roll has fallen from 123mm to 112mm, whilst keeping the amount of paper the same. (I hope no makers were relying on the constant diameter of the correspondingly shrinking tube for any long term projects.)

This (123-112)/123 = 1-0.91 = 9% reduction in diameter doesn't look much for a single roll:

123mm v 112mm, to scale                                                                                      overlaid for comparison

However, once you get a lot of rolls, the saving soon adds up:

the difference is clear with 40 rolls

Assuming that the height of the rolls is the same, the 9% reduction in diameter corresponds to a 1-0.91² = 1-0.83 = 17% reduction in volume.  Hence more rolls can be packed in each delivery lorry, with a claimed  saving of 500 lorry trips, or 140 tonnes of CO₂, per year.

But we could do better, surely?  I've shown the rolls on a square grid, because that's how they are packed:

But what about a hexagonal packing?

smaller rolls, and now hexagonally packed

Now each roll occupies a hexagon of half-height r, not a square of half-height r:

square boundary                                             hexagonal boundary                                             equilateral triangle

What space does this take?  The hexagon is made of 6 equilateral triangles, each of height r.  If the side is of length h, then we have r = h sin 60 = h √3 / 2.  The area of the triangle = 1/2 x base x perpendicular height = h/2 x r = r² / √3.  The area of the hexagon is 6 times this, or  2√3 r²  = 3.464  r².  The square, meanwhile, has area 4 r².  (And not coincidentally, 3.464 is a better approximation to π than is 4.)  Hence hexagonal packing is (4 - 3.464)/4 = 0.134, or a little over 13%, better than square packing.

Doing the sums to combine the percentages properly [viz, 1-(1-17%)(1-13%)=0.28], this means hexagonal packing and smaller rolls combined gives about 28% improvement over the original large, square packed rolls.

So Sainsbury's could save a further 300-odd lorry trips, and a further 90 tonnes CO₂, by packing the rolls hexagonally.

Maybe I should write them a letter?