On May 28, 2015, at 10:36 AM, Warren D Smith <warren.wds@gmail.com> wrote:
Re some earlier posters muttering about "fundamental domains," octagons, and so on, I do not understand what they meant. It seems to me, sqrt(2) is irrational. Therefore, there is no domain tiling the plane by translation, that works for both G1 and G2 simultaneously. And there is no octagon that tiles the plane. And there is no 2D crystal group with 8-fold symmetry.
Take a look at this: http://arxiv.org/pdf/1305.1798.pdf Figure 4 shows such a bijection-generating fundamental domain. The diameter is given in equation (1), which, after you divide by 2, gives the bound 0.92 … on c (for the L_infinity norm). This is actually a very good bound, considering the fact that 1/sqrt(2) = 0.707 is a lower bound. Exercise: Prove that c >= 1/sqrt(2). -Veit