Warren wrote:
Now I remark it also is possible to make a "Goucher lego" not based on a 1x1 square tiling the plane, but rather based on a regular hexagon tiling plane with unit center-separations. The legos will be hexagonal prisms with knobbies on top and indentations on bottom. Then the lattice of "knobbies" is based on "Eisenstein integers" not Gaussian integers. Gaussians: A+Bi, A,B plain-integers, i=squareroot(-1). Eisensteins: A+Bw, A,B plain-integers, w=nontrivial cube root of unity.
Nice. It has occurred to me that my Lego brick is not `primitive', and can be reduced to one where the upper knobbles are rotated by (2+i) and the lower indentations are rotated by (2-i). Similarly, for your hexagon, the upper knobbles can be rotated through (3+omega) and the lower indentations by (3-omega). This avoids the complication of having knobbles halfway between tiles. I shall send pictures shortly. Sincerely, Adam P. Goucher