Another thing that only happens in d > 3 is for the automorphism group to have subgroups whose invariant subspaces are completely irrational. Penrose tilings are non-intuitive mostly for this reason. -Veit
On Jan 14, 2016, at 8:01 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Note the following slightly unintuitive fact:
"There exist full-rank sublattices L of the cubic lattice Z^d such that the point-fixing automorphism group of L is greater than that of Z^d."
For instance, in Z^4 we have that:
L := {(a, b, c, d) : a + b + c + d is even}
has an origin-fixing automorphism group of order 1152 (compared with 384 for the lattice Z^4).
Similarly, in Z^8 we can take a scaled E_8 lattice:
L := {(a_0, a_1, ..., a_7) : all coordinates have the same parity and a_0 + a_1 + ... + a_7 is divisible by 4}
which has an automorphism group of 696729600, compared with 10321920 for the cubic lattice Z^8.
The reason this phenomenon is slightly unintuitive is that it doesn't occur in dimensions d <= 3 so defies our geometric intuition.
Best wishes,
Adam P. Goucher