It must have come as an immense relief to all greengrocers when, in 1998, US mathematician Thomas Hales proved that the way they had been stacking oranges for centuries was in fact the most efficient way possible. Each orange (sphere) in the first layer of such a stack is surrounded by six others to form a hexagonal, honeycomb lattice, while the second layer is built by placing the spheres above the “hollows” in the first layer. The third layer can be placed either directly above the first (producing a hexagonal close-packed lattice structure) or offset by one hollow (producing a face-centred cubic lattice). In both cases, 74% of the total volume of the stack is filled — and Hales showed that this density cannot be bettered.

In the optimal packing arrangement, each sphere is touched by 12 others positioned around it. Newton suspected that this “kissing number” of 12 is the maximum possible in 3D, yet it was not until 1874 that mathematicians proved him right. This is because such a proof must take into account all possible arrangements of spheres, not just regular ones, and for centuries people thought that the extra space or “slop” in the 3D arrangement might allow a 13th sphere to be squeezed in. For similar reasons, Hales’ proof of greengrocers’ everyday experience is so complex that even now the referees are only 99% sure that it is correct.

Greengrocers working in higher-dimensional spaces would find it much harder to stack oranges so efficiently. Generally, the packing density decreases as the number of dimensions increases because the unfilled gaps between the spheres (which themselves are higher dimensional) get bigger. Although mathematicians have worked out the optimal lattice packings for all dimensions up to and including eight, it is not yet known whether there are better irregular arrangements in more than three dimensions. In the 8D case, however, something special happens: the gaps in the array of spheres are just big enough to insert another complete copy of the existing array. The result is a fantastically beautiful 8D lattice called E8, which has a structure analogous to that of diamond but with an even higher degree of symmetry. Each sphere in the E8 lattice is surrounded by 240 others in a tight, slop-free arrangement — solving both the optimal-packing and kissing-number problems in 8D. Moreover, the centres of the spheres mark the vertices of an 8D solid called the E8 or “Gosset” polytope, which is named after the British mathematician Thorold Gosset who discovered it in 1900.

Much of the fascinating symmetry of E8 is squeezed out when it is projected onto a 2D form that we can visualize on paper or on screen (see opposite). But this did not prevent colourful images of the esoteric mathematical object from adorning the front pages of national newspapers last year. In March 2007 a team of about 20 mathematicians from the US and Europe completed a four-year project to map certain properties of E8, which involved calculating the coefficients of over 200 billion lengthy polynomials. That was only the beginning of E8’s fame. A few months later, in a completely independent development, a freelance physicist called Garrett Lisi posted a paper on the arXiv preprint server (“An exceptionally simple theory of everything”) claiming that this same superbly symmetric 8D structure could form the basis of a “theory of everything” that unifies nature’s four known forces: gravity, electromagnetism and the strong and weak nuclear forces. Coupled with Lisi’s unlikely day-job as a surfer based in Hawaii, E8 was all over the newspapers, airwaves and blogs once again.

But what has this oddity of higher-dimensional geometry got to do with physics in the first place? E8 is an example of a symmetry group, the mathematical objects used to describe symmetry. Symmetry principles are deeply rooted in fundamental theoretical physics and have been the driving force behind the hugely successful Standard Model of particle physics. According to Lisi, the beautiful, if complicated, E8 may shed light on how to extend the Standard Model to something closer to a theory of everything.

In the July issue of Physics World, Stephen Maxfield goes on to explain the role of symmetry in physics and asks whether Lisi's claims stand up to scrutiny.

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