The eccentric engineer: solving the Christmas wrapping conundrum

Wrapping presents has a long history and is, so the social anthropologists tell us, a way of “disguising the commodity and adding a layer of authenticity and personal feeling missing from marketplace transactions”. Personally, I use it as a way of camouflaging a disappointing gift for long enough to give me a chance of getting away before the unhappy recipient realises what they’ve got.

And we do love wrapping presents. Each Christmas, we in the UK use enough wrapping paper to cover the whole of the island of Guernsey, which would make a lovely gift. But present wrapping also brings round one of the greatest conundrums in modern engineering. How do you wrap that oddly shaped present?

Hardest of all to wrap are spherical things, thanks to their Gaussian curvature. Flat planes of paper clearly have zero points of curvature – but each time you create a fold in that plane, you add a point and hence get closer to approximating the curvature of the present.

However, to get as much curvature as a sphere, you would have to make an infinite number of folds which, assuming each fold takes some time, would take an infinite amount of time – which wrapping presents seems to do anyway. Don’t forget that wrapping paper can be very expensive, so we want to use it wisely. The most expensive wrapping paper ever was a range of oriental posters advertising the magician Chung Ling Soo (real name Billy Robinson). During paper shortages in the Second World War, they were used to wrap magic tricks – which is why they’re now scarce enough to fetch up to £20,000 each. Billy always kept up his elaborate front about being Chinese and the only time he spoke English on stage was when a ‘catch the bullet’ trick went wrong in 1918. He said: “Oh my God, something’s happened. Lower the curtain,” at which point he promptly died. So, he never got to profit from his wildly expensive wrapping paper.

I digress – back to that awkwardly shaped present. Now there are ‘wrapping gurus’ who claim to be able to help. A quick look online will show they recommend putting strangely shaped objects in a box or bag filled with shredded paper. But where’s the fun in that?

Fortunately, fine engineering and mathematical minds have considered this problem and, thanks to a 19th century Austrian chocolate, come up with something of an answer – foil.

The story begins in 1890, when confectioner Paul Fürst of Salzburg invented a perfectly spherical chocolate, which he named the Mozartkugel after his hometown’s most famous son. Of course, each perfectly spherical confection had to be wrapped and, as Fürst made a lot of Mozartkugel, they had to be wrapped efficiently. The solution he found was to use a square of foil, which could be wrapped around the sweet quickly, using many random crinkles in the surface to approximate the curvature of the contents. To do this, the Mozartkugel was (and still is) placed on a square of foil, the centre of which is on one pole of the sphere, and then the square’s corners are wrapped around the sweet, so they meet at the opposite pole, before finally crinkling the edges. For this method, you need to use a square whose diagonal length is equal to the length of the sphere’s circumference.

Yet is this the most efficient way of wrapping it? Fürst’s more commercial (and less handmade) competitor Mirabell use a rectangle of foil, wrapping the long side of the rectangle around the equator of the sphere, and then squeezing the top and bottom parts to cover the poles. The rectangle must hence be as long as the sphere’s circumference, and half a circumference high.

Which is best? And is there another way? The answer came from MIT computer scientist Erik Demaine and his team who studied both for their seminal paper ‘Wrapping the Mozartkugel’ in Abstracts of the 20th European Workshop on Computational Geometry which was, appropriately enough, held in Austria.

Demaine and his team selflessly bought and carefully unwrapped large numbers of both confections, smoothing out the paper to discover the geometrical secrets of its manufacture. They also felt duty bound to eat the chocolates in the process.

Thanks to some fancy computational geometry, they discovered that there was a way to wrap the chocolates more efficiently by placing the sphere on a foil triangle and wrapping the three corners of the triangle around and slightly past the sphere’s opposite pole. This reduced the wrapping area by a slightly unimpressive 0.1 per cent. However, when you make millions of sweets a year, even those tiny savings can add up.

Why bother? In a footnote to their article, Demaine’s team sagely opine: “The reduced material usage also indirectly cuts down on CO2 emissions, and therefore partially solves the global-warming problem and consequently the little-reported but equally important chocolate-melting problem.”

There are, of course, other solutions. Demaine suggests one made of ‘petals’ of foil, rather like the gores of a map, which could save 20 per cent, but would require cutting a complex shape and working out how best to make the shape tile the plane so it could be repeatedly and efficiently cut out from a single sheet of foil. Then it would also have to be carefully wrapped around the chocolate. At this point, everyone just decided to have another chocolate and forget it.