After some considerable fighting I managed to obtain a few rules to generate tilings of the cubic 3D grid with tiles that are up to (one) rotation and shifts copies of themselves. (Think of Gosper's island in 3D). Terminology: is "rep-tile" the proper term for these things? Here is one such rule. Let {D = [ [ 0, 0, 0]~, [ 0, 0, 2]~, [ 0, 1, 1]~, [ 0, 1, -1]~, [ 1, 1, 0]~, [ 1, 1, 2]~, [ 1, 2, 1]~, [ 1, 2, -1]~ ]; } (this is a little parallelepiped out of 8 points, only even coordinates are used). Set S = [ [0,0,0]~ ]; (just the origin). Now we need a matrix M with determinant 8. Repeatedly set S = M * (s - d) for all s \in S and d \in D Our set S grows by a factor of 8 with each such step. As a Pari fragment: map( S, D, M )= { my( T=vector(#D * #S), ct=0 ); for (j=1, #D, my(d=D[j]); for (k=1, #S, my(s=S[k]); ct += 1; T[ct] = M * (s - d); ); ); return( T ); } Using M = 2 * (identity matrix) gives bigger and bigger parallelepipeds (not really a surprise). Now set { M = 2 * [ 0, 0, 1; 1, 0, 0; 0, 1, 0]; } (rotation by 120 degrees around the vector [1, 1, 1]) and you DO get a surprise, a very nontrivial 3D tile. I can only render the thing as point cloud and so have no useful images to offer. Maybe some computerhead on this list can do better. I can email the point cloud (full or just the hull) if anybody wants the data. Apparently the problem "point cloud --> closed shape" is nontrivial, and made even harder by the fractal-ish nature of the surfaces we have here. If the rendering turns out to be too nasty I can give a systems with card(D) = 27 that at least as a point cloud looks much easier to grasp. Bonus track: Three more variations. Do the rotation by 60 degrees. Do both above with the following (compacted) parallelepiped {D = [ [ 0, 0, 0]~, [ 1, 0, 0]~, [ 0, 1, 0]~, [ 1, 1, 0]~, [ 1, 1, 1]~, [ 1, 2, 1]~, [ 2, 1, 1]~, [ 2, 2, 1]~ ]; } Best regards, jj