There are some striking periodicities in the table above. Modulo 2 we see that going
from k= 2nto k= 2n+ 1 we alternately add Cor ˜
C, which depending on if we are
looking at the CL(0, k)column, or CL(k , 0). And modulo 4 we see that
CL(0,4n)' CL(4n, 0), n ≥0.
Modulo 8 is the big periodicity, related to what is known as Bott periodicity. In this
context we ﬁrst see that at k= 8 there is a kind of algebraic collapse, or simpliﬁcation,
in the representation. But also,
CL(0, k + 8) ' CL(0, k)⊗ CL(0,8),
CL(k+ 8,0) ' CL(k, 0) ⊗ CL(0,8).
This kind of order 8 periodicity applies as well to CL(p, q ), with neither pnor qequal
to 0, but I’m not interested in that here. However, in that case we lose the split duality.
For example, CL(1,1) '˜
HL, which is not self-dual.
Lattices and Dimensions 1, 2, 8 and 24
I accumulated most of my ideas (and what I know) about lattice theory and sphere
packings in . My interest in the Leech lattice, speciﬁcally, derives from , and it
relates to my investigations into the roles exceptional mathematical objects, like the
division algebras, play in theoretical physics (, , ). Conway and Sloane make
it abundantly apparent that the Leech lattice satisﬁes a great many criteria for excep-
tionality in this notoriously complex ﬁeld.
One of the leaders in the ﬁeld is Henry Cohn who wrote a paper summarizing a
recent breakthrough . I’d like to share a few quotes. The initial breakthrough, the
work of M. S. Viazovska, related to the sphere packing problem in 8 dimensions .
She proved that E8(8-dimensional laminated lattice, also denoted Λ8) is the densest
sphere packing in 8 dimensions. Cohn says:
No proof of optimality had been known for any dimension above three, and
Viazovskas paper does not even address four through seven dimensions.
Cohn and collaborators then applied Viazovskas method to prove the Leech lattice
(Λ24) is the desnsest packing in 24 dimensions. And again, their work skirts all the
intermediate dimensions, 9 to 23. Cohn says:
Unfortunately, our low-dimensional experience is poor preparation for un-
derstanding high-dimensional sphere packing. Based on the ﬁrst three di-
mensions, it appears that guessing the optimal packing is easy, but this
expectation turns out to be completely false in high dimensions.
The sphere packing problem seems to have no simple, systematic solution
that works across all dimensions. Instead, each dimension has its own id-
iosyncracies and charm. Understanding the densest sphere packing in R8
tells us only a little about R7or R9, and hardly anything about R10.