Warren wrote:

but I didnt think the Taylor-Soc tile was really a tile. It used color matching rules, not real tile rules based only on shape. No?

There are three different variants: I: A connected two-dimensional tile with colour-matching rules; II: A disconnected tile where matching is forced geometrically; III: A complicated three-dimensional connected tile. Now, with access to the third dimension it is straightforward to implement II as a connected prismatic hexagonal tile (with lots of interlocking corkscrew-like protrusions). This can be embedded in the turnover version of the hexagonal Lego tile as described in the previous e-mail.

I thought of the checkerboard of indent+knobs idea too allowing turnover too, great minds think alike :). What I'd really like to solve is the einstein problem in 2D or 3D, though.

The strongly aperiodic version (no infinite cylic group of symmetries), I guess you mean...? The weakly aperiodic (no translations) has of course already been solved by the Schmitt tile, SCD tile or Goucher lego (whichever you prefer).

You seem to be very smart and you know about the right stuff so you might be able to.

Thanks.

I think it is going to take conceptual clarifications aimed at figuring out how to make a computer explore the possibilities efficiently.

Yes. In the same way that the Taylor-Socolar tile is a decorated hexagon, I suspect that there is a decorated rhombic dodecahedron or decorated truncated octahedron capable of tiling space strongly aperiodically. As long as you don't allow both enantiomorphs of a tile, you can replace colour-matching rules with indentations and protrustions. Sincerely, Adam P. Goucher