For the general public let me just expand a drop. The lattice gives the best packing in D=8 [E(8)]. And also there are other important things about D=8. We get the right relationships between vector particles and spinors. We get the right number of fermions.
And plus that E(8) is reducible to d=3 and that is nice for a 3d world.
See this paragraph written by Warren Siegel:
From Warren Siegel:
However, these spinors can have the usual commutation relations and conformal weights only for D=8. This is significant for two reasons: (1) D=8 is the number of physical (i.e., transverse) fermions for the RNS superstring, and (2) SO(8) is the only simple Lie group with the property of “triality”, a symmetry between the vector and two spinor representations. In fact, if we start out by defining the basis for one of the spinors with the same a we used above to define the vector, and rewrite the above a’s for the vector and other spinor in terms of that new basis, we see that we have just permuted the 3 a’s. [Fields pg 776]