A student of Imperial University consulted me about a project in which multiple transmutation engines ran in parallel, operating identical designs and sharing internal components below the interface layer of elemental proxies and gripper arms. A strange issue in his project emerged: the timing crystal in each transmutation engine, which normally keeps its arms operating in lockstep on the instruction tape, would become slightly out-of-phase with every other timing crystal, leading to cascading conflicts and an eventual "Arms cannot operate asynchronously" fatal error, which I had never seen before.

These phase drifts are small, and a timing crystal's full oscillation is one loop through the instruction tape. Therefore, on short tapes, each instruction spans a wider range of phases than on long tapes. To change the transmutation engines' interconnections would cause more problems than it solved, so his problem was reduced to this--within these tesselating super-hexagons, minimize the length of the instruction tapes, i.e. period. 

Except, no, because "period" CHAFES against the constraint of hexagons! There is no room to sprawl, no place to scribble track, and hardly a place to put the output glyph, even! There is a good reason that the student found little prior art for his project.

This mirror coat formula demonstrates the issue. Typically, a reasonable minimum period to attempt is 4, and an unreasonable minimum is 3. This machine leans heavily on the leeway of period 5 for almost every arm. I'm confident that period 4 will be found and re-found, but not by me. 3 would be astonishing. I asked my correspondent for help, and we were both stumped for improvements.

Best of luck to those tinkering under the surface in Imperial University. Strange restrictions, far beyond price, speed and size, suddenly become deeply practical constraints.

- Alchemist Kazyan, in correspondence with Alchemist Jasper