Warren Smith wrote:
The hex-lego has no "slide the strips" freedom unlike the square-lego -- its layers thus are unique. It might be that some combinations of upper & lower twists eliminate the latter freedom though.
The square `Goucher lego' doesn't have `slide the strips' freedom, either, since the knobbles only line up to form a square lattice if the tiles themselves form a square lattice. Animation: http://cp4space.files.wordpress.com/2013/08/noslide.gif Anyway, I've made a hexagonal monotile based on your suggestion: http://cp4space.files.wordpress.com/2013/08/hex-monotiles.png The indentations in the bottom are rotated by the same angle as the knobbles on the top (but in opposite directions). It's possible to replace both the indentations and knobbles with a _self-complementary_ ring of alternate indentations and knobbles. The advantage is that the tile is now invariant under being flipped over. This means that we can embed a 2D hexagonal tile in the equator of the lego, where both enantiomers of the 2D tile are allowed to appear in a layer (since we can flip the 3D tile over). In particular, choose the Taylor-Socolar tile, forcing an aperiodic tiling in each layer. The result is a 3D tile with the following pair of properties: (a) Imitates the Taylor-Socolar pattern in each layer; (b) Has no translational symmetry (being instead screw-symmetric). This is, in a sense, the most aperiodic 3D tile discovered thus far. Sincerely, Adam P. Goucher