University of Warwick mathematician Ian Stewart has provided New Scientist with a scientific guide to gift wrapping. Very festive. Professor Stewart informs us about the “sausage conjecture”, which asks what the most efficient way to wrap a group of circles or spheres is.
For two and three dimensions, we have the answer: round circular objects (like mince pies) should be stacked end to end like a sausage if you have six or fewer, but for seven or more you’re better off arranging the pies in a hexagon and wrapping them that way in order to minimise the paper used. For spherical objects (Christmas puddings, of course) the split comes at 56 or fewer versus 57 or more.
So far, so simple, and good enough for anyone looking to wrap presents this Christmas. You might think we could just leave it there, but mathematicians never can. Extend the problem to four dimensions, and matters become predictably more complex. Now, you might be asking what a four-dimensional sphere looks like, and the truth is it’s impossible for the human mind to visualise. Mathematicians have no trouble with higher dimensions however – just add another number to your coordinate system. So, whilst we need two numbers to describe any point on a circle, and three numbers for a sphere, a group of four numbers will let us mathematically explore a so-called hypersphere.
How exactly do you go about wrapping a group of hyperspheres them? Well, for 50,000 or fewer you’re looking at a hyper-sausage, and for 100,000 you’re looking at something distinctly un-sausage-like – thought no-one knows exactly what. As for the specific trade off point, it isn’t as clear cut as with circles or spheres, but it definitely lies between 50,000 and 100,000 hyperspheres.
So what about the “sausage conjecture”? Unfortunately, it’s nothing to do with the trimmings at Christmas dinner, but rather states that for objects with five-dimensions or more, sausages are always best. This rather uninituive result, given the rules for two, three and four dimensions, was put forward in 1975 by Hungarian mathematician László Fejes Tóth.
Whilst it’s no Fermat’s Last Theorem or Riemann Hypothesis, some headway has been made with the sausage conjecture. In 1998 Ulrich Betke, Martin Henk and Jörg Wills proved that it was true for 42 or more dimensions, just leaving the cases 5 to 41. Perhaps you’d like to contemplate them as you wrap your Christmas presents!