Actually, that lego is quite strange. If you just take 2 "layers" of the tiling that IS periodic (viewed 2-dimensionally)! Indeed ANY fixed finite number L of "layers" is something that is periodic viewed 2-dimensionally! Proof sketch: It's because ((3+4i)/5)^L is a plain-integer scaling of a Gaussian integer, if L>=0 is any plain integer. In that respect the "Goucher lego" is different than the Conway-Schmitt tiles (or than almost all of them, anyhow).