Goucher's lego is based on a 1x1 square. We could equally well base it on a 2x1 rectangle (two glued together). Also, Goucher's lego as originally described actually will tile in more than one way. If you have 1x1 squares you can tile the plane in a non-unique way, make 1Xinfinity vertical strips and each strip can be slid vertically. Now for Goucher's lego, you can slide the strips certain discrete amounts within each layer, so there are an infinity of different ways to tile 3-space (or merely a finite number of layers) with it. However one could, if desired, slightly perturb the shape (e.g. the 1x1 square becomes a shape close to a 1x1 square, but with slightly S-curved edges) to force uniqueness of one of these ways. 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. Both Gaussians and Eisensteins enjoy unique factorization. Now the magic choice Goucher made based on 3^2 + 4^2 = 5^2 [or any primitive Pythagorean triple (5,12,13), (8,15,17) etc], is replaced by 3^2 - 3*8 + 8^2 = 7^2 [or (5,8,7) or (7,15,13) or (16,21,19) etc; if (a,b,c) works so does (b-a,b,c)]. That is, the magic rotation of the knobbie lattice is the unit-norm complex number (3+8w)/7 instead of (3+4i)/5. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)