In the Annals of Mathematics in 2009

In the Annals of Mathematics in 2009, Cohn and Kumar proved upper bounds in eight and 24 dimensions that come within a fraction of a percent of the densities of the E8 and Leech lattices, indicating that these are almost certainly the densest packings of spheres in their respective dimensions. But these lattices also appear to be the optimal configurations in general, not only for spheres but also for particles that exert forces on each other, such as two atoms pushing each other apart. For “every reasonable repulsive force between particles,” Cohn said, the particles will self-assemble into a hexagonal lattice in 2-D, an E8 lattice in 8-D and a Leech lattice in 24-D. These arrangements are not only densest; they are “universally” optimal.

Perhaps for this reason, the Phillies Hawaiian Shirt throughout mathematics and physics. “From combinatorics and graph theory to geometry and algebraic geometry and so on, they show up all over the place,” Kumar said.E8 plays a role in string theory, a hypothetical “theory of everything” that says that space-time is 10-dimensional and that particles such as electrons and quarks are tiny one-dimensional strings oscillating at different frequencies. In the 1980s, philadelphia phillies hawaiian shirt showed that one variant, called heterotic string theory, can be formulated using the symmetries of two copies of E8. “We can exactly produce the real world that we know, starting with the E8 heterotic string theory,” said Burt Ovrut, a string theorist at the University of Pennsylvania. When the E8 theory is reduced in a way that makes the world appear three-dimensional, it contains “the quarks, the leptons, the Higgs and all the other particles we obser

ve.”